1MA 02. RESOLUCIÓ DE TRIANGLES/1MA.02.1 Resolució de triangles rectangles/1MA.02.1.1 Resolució segons els costats i//o l'angle coneguts 1MA.02.1.1.10 DT PITÀGORES RAONS

Teorema de Pitàgores

1. Coneguts b i c, s'aplica el teorema de Pitàgores per trobar la hipotenusa a: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»a«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»=«/mo»«msqrt mathcolor=¨#00007F¨»«msup»«mi mathvariant=¨bold¨»b«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»+«/mo»«msup»«mi mathvariant=¨bold¨»c«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/msqrt»«/mrow»«/mstyle»«/math».

2. Si es coneix a i b (o a i c) es calcula el costat que falta amb «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»c«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»=«/mo»«msqrt mathcolor=¨#00007F¨»«msup»«mi mathvariant=¨bold¨»a«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»-«/mo»«msup»«mi mathvariant=¨bold¨»b«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/msqrt»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»o«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»b«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»=«/mo»«msqrt mathcolor=¨#00007F¨»«msup»«mi mathvariant=¨bold¨»a«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»-«/mo»«msup»«mi mathvariant=¨bold¨»c«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/msqrt»«/mrow»«/mstyle»«/math»

3. Per determinar els angles aguts, es fan servir les raons trigonomètriques: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨ mathsize=¨12px¨»B«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨ mathsize=¨12px¨»=«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨ mathsize=¨12px¨»a«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨ mathsize=¨12px¨»s«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨ mathsize=¨12px¨»i«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨ mathsize=¨12px¨»n«/mi»«mfrac mathcolor=¨#00007F¨»«mi mathvariant=¨bold¨ mathsize=¨12px¨»b«/mi»«mi mathvariant=¨bold¨ mathsize=¨12px¨»a«/mi»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨ mathsize=¨12px¨»;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨ mathsize=¨12px¨»§#8201;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨ mathsize=¨12px¨»C«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨ mathsize=¨12px¨»=«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨ mathsize=¨12px¨»a«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨ mathsize=¨12px¨»s«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨ mathsize=¨12px¨»i«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨ mathsize=¨12px¨»n«/mi»«mfrac mathcolor=¨#00007F¨»«mi mathvariant=¨bold¨ mathsize=¨12px¨»c«/mi»«mi mathvariant=¨bold¨ mathsize=¨12px¨»a«/mi»«/mfrac»«mstyle mathsize=¨12px¨/»«/math»

Raons trigonomètriques

sin B = b/a; cos B = c/a; tg B = b/c

]]> 0.0000000 0.0000000 0 1MA.02.1.1.11Q Resolució(b,c → a,A,B,C) En un triangle rectangle, els costats b i c mesuren, respectivament, «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»b«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»i«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»c«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«/mstyle»«/math» m.
Resol el triangle.

Format: arrodonit (a la unitat). Angles en graus sense unitat

]]> 1.0000000 0.2500000 0 0 a =#sol1B =#sol2C=#sol3]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>11</mn><mo>,</mo><mn>99</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>11</mn><mo>,</mo><mn>79</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><msqrt><mrow><msup><mi>b1</mi><mn>2</mn></msup><mo>+</mo><msup><mi>c1</mi><mn>2</mn></msup></mrow></msqrt></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi><mo>=</mo><mi>arrodoneix</mi><mo>(</mo><mi>a</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>=</mo><mfrac><mn>180</mn><pi/></mfrac><mo>*</mo><mi>asin</mi><mfenced><mfrac><mi>b1</mi><mi>a</mi></mfrac></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi><mo>=</mo><mi>arrodoneix</mi><mo>(</mo><mi>B</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>=</mo><mn>90</mn><mo>-</mo><mi>B</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol3</mi><mo>=</mo><mi>arrodoneix</mi><mo>(</mo><mi>C</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b1</mi><mo>,</mo><mi>c1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>65</mn><mo>,</mo><mn>30</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>*</mo><msqrt><mn>205</mn></msqrt></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>72</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>65.225</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>65</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>24.775</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol3</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>25</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>1</mn><mspace linebreak="newline"/><mi>B</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>2</mn><mspace linebreak="newline"/><mi>C</mi><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>3</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-4)</option><option name="relative_tolerance">false</option><option name="precision">4</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputCompound">true</data><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> a es calcula amb el teorema de Pitàgores

B és l'arc sinus de #b/#a (cal fer-ho amb la hipotenusa sense arrodonir)

Per a trobar C, com que C + B = 90º, fem la resta 90º - #B.

]]> 1MA.02.1.1.12Q Resol(b,B → a,c,C). En un triangle rectangle, l'angle B mesura «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»B_«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#186;«/mo»«/mstyle»«/math» i el costat b mesura «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»b_«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«/mstyle»«/math» m.
Resol el triangle.

Escriu els resultats arrodonits.

]]> 1.0000000 0.2500000 0 0 C =#C_1a =#a_1c =#c_1]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>11</mn><mo>,</mo><mn>999</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>11</mn><mo>,</mo><mn>79</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C_1</mi><mo>=</mo><mn>90</mn><mo>-</mo><mi>B_1</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mfrac><mrow><mi>B_1</mi><mo>*</mo><pi/></mrow><mn>180</mn></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_1</mi><mo>=</mo><mi>arrodonir</mi><mfenced><mrow><mfrac><mi>b_1</mi><mrow><mi>sin</mi><mo>(</mo><mi>r</mi><mo>)</mo></mrow></mfrac><mo>*</mo><mn>1</mn><mo>.</mo><mn>0</mn></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_1</mi><mo>=</mo><mi>arrodonir</mi><mfenced><mrow><mfrac><mi>b_1</mi><mrow><mi>tan</mi><mo>(</mo><mi>r</mi><mo>)</mo></mrow></mfrac><mo>*</mo><mn>1</mn><mo>.</mo><mn>0</mn></mrow></mfenced></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>974</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>76</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>14</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1004</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>243</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>C</mi><mi>_</mi><mn>1</mn><mspace linebreak="newline"/><mi>a</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>a</mi><mi>_</mi><mn>1</mn><mspace linebreak="newline"/><mi>c</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>c</mi><mi>_</mi><mn>1</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-4)</option><option name="relative_tolerance">false</option><option name="precision">4</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputCompound">true</data><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> Per a trobar C, com que C + B = 90º, fem la resta 90º - #B_1.

]]> Per calcular a (hipotenusa) a partir de l'oposat b = #b_1, utilitzem el sinus de B:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨bold¨ mathcolor=¨#007F00¨»sin«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#007F00¨»B«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»=«/mo»«mfrac mathcolor=¨#007F00¨»«mi mathvariant=¨bold¨»b«/mi»«mi mathvariant=¨bold¨»a«/mi»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#8660;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#007F00¨»a«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mfrac mathcolor=¨#007F00¨»«mi mathvariant=¨bold¨»b«/mi»«mrow»«mi mathvariant=¨bold¨»sin«/mi»«mi mathvariant=¨bold¨»B«/mi»«/mrow»«/mfrac»«/math»

]]> Per calcular c (adjacent) a partir de l'oposat b = #b_1, utilitzem la tangent de B:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#007F00¨»t«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#007F00¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#007F00¨»B«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#8201;«/mo»«mfrac mathcolor=¨#007F00¨»«mi mathvariant=¨bold¨»b«/mi»«mi mathvariant=¨bold¨»c«/mi»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#8660;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#007F00¨»c«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mfrac mathcolor=¨#007F00¨»«mi mathvariant=¨bold¨»b«/mi»«mrow»«mi mathvariant=¨bold¨»tg«/mi»«mo mathvariant=¨bold¨»§#160;«/mo»«mi mathvariant=¨bold¨»B«/mi»«/mrow»«/mfrac»«/math»

]]> 1MA.02.1.1.13Q Resol(c,B → a,b,C). En un triangle rectangle, l'angle B mesura «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»B_«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#186;«/mo»«/mstyle»«/math» i el costat c mesura «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»c_«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«/mstyle»«/math» m.
Resol el triangle.

Escriviu els resultats arrodonits.

]]> 1.0000000 0.2500000 0 0 C =#C_1a =#a_1b =#b_1]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>11</mn><mo>,</mo><mn>999</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>11</mn><mo>,</mo><mn>79</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C_1</mi><mo>=</mo><mn>90</mn><mo>-</mo><mi>B_1</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mfrac><mrow><mi>B_1</mi><mo>*</mo><pi/></mrow><mn>180</mn></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_1</mi><mo>=</mo><mi>arrodonir</mi><mfenced><mrow><mfrac><mi>c_1</mi><mrow><mi>cos</mi><mo>(</mo><mi>r</mi><mo>)</mo></mrow></mfrac><mo>*</mo><mn>1</mn><mo>.</mo><mn>0</mn></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b_1</mi><mo>=</mo><mi>arrodonir</mi><mfenced><mrow><mi>c_1</mi><mo>*</mo><mi>tan</mi><mo>(</mo><mi>r</mi><mo>)</mo><mo>*</mo><mn>1</mn><mo>.</mo><mn>0</mn></mrow></mfenced></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>727</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>48</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>42</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1086</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>807</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>C</mi><mi>_</mi><mn>1</mn><mspace linebreak="newline"/><mi>a</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>a</mi><mi>_</mi><mn>1</mn><mspace linebreak="newline"/><mi>b</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>b</mi><mi>_</mi><mn>1</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-4)</option><option name="relative_tolerance">false</option><option name="precision">4</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputCompound">true</data><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> Per a trobar C, com que C + B = 90º, fem la resta 90º - #B_1.

]]> Per calcular a (hipotenusa) a partir de l'adjacent c = #c_1, utilitzem el cosinus de B:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨bold¨ mathcolor=¨#007F00¨»cos«/mi»«mo»§#160;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#007F00¨»B«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»=«/mo»«mfrac mathcolor=¨#007F00¨»«mi mathvariant=¨bold¨»c«/mi»«mi mathvariant=¨bold¨»a«/mi»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#8660;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#007F00¨»a«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mfrac mathcolor=¨#007F00¨»«mi mathvariant=¨bold¨»c«/mi»«mrow»«mi mathvariant=¨bold¨»cos«/mi»«mo»§#160;«/mo»«mi mathvariant=¨bold¨»B«/mi»«/mrow»«/mfrac»«/math»

]]> Per calcular b (oposat) a partir de l'adjacent c = #c_1, utilitzem la tangent de B:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#007F00¨»t«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#007F00¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#007F00¨»B«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#8201;«/mo»«mfrac mathcolor=¨#007F00¨»«mi mathvariant=¨bold¨»b«/mi»«mi mathvariant=¨bold¨»c«/mi»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#8660;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#007F00¨»b«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#007F00¨»c«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#183;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#007F00¨»t«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#007F00¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#007F00¨»B«/mi»«/math»

]]> 1MA.02.1.1.14Q Resol(a,B → b,c,C) En un triangle rectangle, l'angle B mesura «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»b«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#186;«/mo»«/mstyle»«/math» i la hipotenusa mesura «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»a«/mi»«/mstyle»«/math» m.
Resol el triangle.

Escriviu els resultats arrodonits.

]]> 1.0000000 0.2500000 0 0 C =#c_1b =#b_1c =#c_2]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>11</mn><mo>,</mo><mn>999</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>11</mn><mo>,</mo><mn>79</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_1</mi><mo>=</mo><mn>90</mn><mo>-</mo><mi>b</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mfrac><mrow><mi>b</mi><mo>*</mo><pi/></mrow><mn>180</mn></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b_1</mi><mo>=</mo><mi>arrodonir</mi><mfenced><mrow><mi>a</mi><mo>*</mo><mi>sin</mi><mo>(</mo><mi>r</mi><mo>)</mo><mo>*</mo><mn>1</mn><mo>.</mo><mn>0</mn></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_2</mi><mo>=</mo><mi>arrodonir</mi><mfenced><mrow><mi>a</mi><mo>*</mo><mi>cos</mi><mo>(</mo><mi>r</mi><mo>)</mo><mo>*</mo><mn>1</mn><mo>.</mo><mn>0</mn></mrow></mfenced></math></input></command></group></library></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>c</mi><mi>_</mi><mn>1</mn><mspace linebreak="newline"/><mi>b</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>b</mi><mi>_</mi><mn>1</mn><mspace linebreak="newline"/><mi>c</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>c</mi><mi>_</mi><mn>2</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-4)</option><option name="relative_tolerance">false</option><option name="precision">4</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputCompound">true</data><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> Per a trobar C, com que C + B = 90º, fem la resta 90º - #B.

]]> Per calcular b (oposat) a partir de la hipotenusa #a, utilitzem el sinus de B:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨bold¨ mathcolor=¨#007F00¨»sin«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#007F00¨»B«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»=«/mo»«mfrac mathcolor=¨#007F00¨»«mi mathvariant=¨bold¨»b«/mi»«mi mathvariant=¨bold¨»a«/mi»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#8660;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#007F00¨»b«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#007F00¨»a«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#007F00¨»sin«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#8201;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#8201;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#007F00¨»B«/mi»«/math»

]]> Per calcular c (adjacent) a partir de la hipotenusa a, utilitzem el cosinus de B:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨bold¨ mathcolor=¨#007F00¨»cos«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#007F00¨»B«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#8201;«/mo»«mfrac mathcolor=¨#007F00¨»«mi mathvariant=¨bold¨»c«/mi»«mi mathvariant=¨bold¨»a«/mi»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#8660;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#007F00¨»c«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#007F00¨»a«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#007F00¨»cos«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#007F00¨»B«/mi»«/math»

]]> 1MA.02.1.1.15Q Resol(a,C → b,c, B) En un triangle rectangle, l'angle C mesura «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»C_«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#186;«/mo»«/mstyle»«/math» i la hipotenusa mesura «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»a«/mi»«/mstyle»«/math» m.
Resol el triangle.

Escriviu els resultats arrodonits.

]]> 1.0000000 0.2500000 0 0 B =#B_1b =#b_1c =#c_1]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>11</mn><mo>,</mo><mn>999</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>11</mn><mo>,</mo><mn>79</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B_1</mi><mo>=</mo><mn>90</mn><mo>-</mo><mi>C_1</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mfrac><mrow><mi>B_1</mi><mo>*</mo><pi/></mrow><mn>180</mn></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b_1</mi><mo>=</mo><mi>arrodonir</mi><mfenced><mrow><mi>a</mi><mo>*</mo><mi>sin</mi><mo>(</mo><mi>r</mi><mo>)</mo><mo>*</mo><mn>1</mn><mo>.</mo><mn>0</mn></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_1</mi><mo>=</mo><mi>arrodonir</mi><mfenced><mrow><mi>a</mi><mo>*</mo><mi>cos</mi><mo>(</mo><mi>r</mi><mo>)</mo><mo>*</mo><mn>1</mn><mo>.</mo><mn>0</mn></mrow></mfenced></math></input></command></group></library></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>B</mi><mi>_</mi><mn>1</mn><mspace linebreak="newline"/><mi>b</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>b</mi><mi>_</mi><mn>1</mn><mspace linebreak="newline"/><mi>c</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>c</mi><mi>_</mi><mn>1</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-4)</option><option name="relative_tolerance">false</option><option name="precision">4</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputCompound">true</data><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> Per a trobar B, com que C + B = 90º, fem la resta 90º - #C_1.

]]> Per calcular b (oposat) a partir de la hipotenusa #a, utilitzem el sinus de B:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨bold¨ mathcolor=¨#007F00¨»sin«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#007F00¨»B«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»=«/mo»«mfrac mathcolor=¨#007F00¨»«mi mathvariant=¨bold¨»b«/mi»«mi mathvariant=¨bold¨»a«/mi»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#8660;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#007F00¨»b«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#007F00¨»a«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#007F00¨»sin«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#8201;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#8201;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#007F00¨»B«/mi»«/math»

]]> Per calcular c (adjacent) a partir de la hipotenusa a, utilitzem el cosinus de B:

]]> 1MA.02.1.1.51Q Pitàgores en hexàgon Calculeu l'àrea d'un hexàgon inscrit en una circumferència de radi #r.

Format de la resposta: arrel simplificada

Ja pots calcular l'àrea del triangle vermell BCE.

]]> Si el triangle vermell té una àrea de #A2, quants triangles vermells formen l'hexàgon?

]]> 1MA 02. RESOLUCIÓ DE TRIANGLES/1MA.02.1 Resolució de triangles rectangles/1MA.02.1.2 Problemes que fan servir les raons 1MA.02.1.2.11Q Tangent(AlçadaEdificiFont) Des del cim d'un edifici, es veu una font situada a #c m del peu de l'edifici sota un angle de #C º. Quina és l'alçada de l'edifici?

Dona el resultat arrodonit a les unitats, sense unitats

]]> 1.0000000 0.5000000 0 0 #r_63 <question><wirisCasSession><session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>11</mn><mo>,</mo><mn>89</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>11</mn><mo>,</mo><mn>999</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mi>C</mi><mo>*</mo><mfrac><pi/><mn>180</mn></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r_63</mi><mo>=</mo><mi>arrodonir</mi><mfenced><mfrac><mi>c</mi><mrow><mi>tan</mi><mo>(</mo><mi>C</mi><csymbol definitionURL="http://.../units/degree/angular">º</csymbol><mo>)</mo></mrow></mfrac></mfenced></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>C</mi><mo>,</mo><mi>c</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>65</mn><mo>,</mo><mn>823</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>13</mn><mo>*</mo><pi/></mrow><mn>36</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r_63</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>384</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><correctAnswers><correctAnswer type="mathml">#r_63</correctAnswer></correctAnswers><localData><data name="inputField">textField</data></localData></question> Cal dibuixar el triangle on l'angle C és #C º i el costat adjacent és #c.

L'angle B = 90º - #C.

I el que volem calcular és el costat oposat a B. Per a trobar el costat oposat, n'hi ha prou amb utilitzar la tangent de B

]]> 1MA.02.1.2.12Q Tangent(VaixellPenyaSegat) Des d'un vaixell situat a «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»c«/mi»«/mstyle»«/math» m de la costa, es veu el cim d'un penya-segat sota un angle de «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»B_«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#00007F¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#186;«/mo»«/mrow»«/mstyle»«/math». Quina és l'alçada del penya-segat?

Format de la resposta: arrodonida al centèsims

]]> 1.0000000 0.5000000 0 0 #w_1]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>25</mn><mo>,</mo><mn>65</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>11</mn><mo>,</mo><mn>35</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mfrac><mrow><mi>B_1</mi><mo>*</mo><pi/></mrow><mn>180</mn></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w_1</mi><mo>=</mo><mi>c</mi><mo>*</mo><mi>tan</mi><mo>(</mo><mi>r</mi><mo>)</mo><mo>*</mo><mn>1</mn><mo>.</mo><mn>0</mn></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tolerancia</mi><mo>(</mo><mn>0</mn><mo>.</mo><mn>01</mn><mo>)</mo></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>40</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>21</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mo>*</mo><pi/></mrow><mn>9</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>17.621</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>#</mo><mi>w</mi><mo>_</mo><mn>1</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="precision">8</option><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">textField</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> Si dibuixes el triangle, la base és el costat c = #c, el vaixell està situat en el vèrtex de l'angle B = #B_1, i ens demanen el costat b, que és l'oposat a l'angle B.

Has d'utilitzar la tangent que relaciona l'oposat (b) i l'adjacent (c).

]]> 1MA.02.1.2.13Q Cosinus(EscalaParet) Una escala està recolzada a una paret. Del peu de la paret a l'escala hi ha «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»c«/mi»«/mrow»«/mstyle»«/math» metres. Quina és la longitud de l'escala si l'angle que forma amb el terra és de «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»B«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#186;«/mo»«/mstyle»«/math»?

Format de la resposta: arrodonida al centèsims

S'escriu que cos #B = #c/a

i per tant, a = #c/cos#B

]]> 1MA.02.1.2.14Q Sinus(ArbreTrencat) Un arbre ha quedat trencat pel vent. Ara la part trencada forma un angle de #Cº amb la part que ha quedat dreta. Quina era l'altura total de l'arbre, si la part que ha quedat dreta mesura #b m?

Format de la resposta: arrodonida al centèsims

S'escriu que cos #C = #b/a

i per tant, a = #b/cos#C.

Després, cal sumar #a i #b per saber l'altura total de l'arbre.

]]> 1MA.02.1.2.15Q Tangent(TorreOmbra) Quina és l'altura d'una torre que projecta una ombra de «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»b«/mi»«mo»§#160;«/mo»«/mrow»«/mstyle»«/math» m quan l'angle que formen els rajos del sol amb el terra és de «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»B«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#176;«/mo»«/mrow»«/mstyle»«/math» .

Format: als mil·lèsims i sense unitats

]]> 1MA.02.1.2.16Q Sinus(EstelCorda) Fem volar un estel que té un cordill de «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»a«/mi»«/mrow»«/mstyle»«/math» m. Quan l’angle que forma amb el terra és de «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»B«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#186;«/mo»«/mstyle»«/math», a quina alçada es troba?

Format: arrodonit als mil·lèsims i sense unitats

]]> 1.0000000 1.0000000 0 0 #sol <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>50</mn><mo>,</mo><mn>100</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>15</mn><mo>,</mo><mn>45</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi><mo>=</mo><mi>a</mi><mo>*</mo><mi>sin</mi><mo>(</mo><mfrac><mrow><mi>B</mi><mo>*</mo><pi/></mrow><mrow><mn>180</mn><mo>.</mo></mrow></mfrac><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tolerància</mi><mo>(</mo><mn>0</mn><mo>.</mo><mn>001</mn><mo>)</mo></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>71</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>50.205</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer>#sol</correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="precision">8</option><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">textField</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> El cordill és la hipotenusa, i l'alçada el costat oposat a l'angle. Escriu el sinus i aïlla l'oposat.

]]> 1MA.02.1.2.21Q RadiPolígonRegular(Costat) Quin és radi d'un polígon regular de «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»n«/mi»«/mrow»«/mstyle»«/math» costats si el costat fa «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»c«/mi»«/mrow»«/mstyle»«/math» cm?

Format: als mil·lèsims i sense unitat

#g

]]> 1.0000000 0.5000000 0 0 #r <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>5</mn><mo>,</mo><mn>12</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>2</mn><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>4</mn><mo>,</mo><mn>6</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>180</mn><mo>º</mo><mo>/</mo><mi>n</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mfrac><mi>c</mi><mrow><mn>2</mn><mo>*</mo><mi>sin</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow></mfrac><mo>*</mo><mn>1</mn><mo>.</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ap</mi><mo>=</mo><msqrt><mrow><msup><mi>r</mi><mn>2</mn></msup><mo>-</mo><mo>(</mo><mi>c</mi><mo>/</mo><mn>2</mn><msup><mo>)</mo><mn>2</mn></msup></mrow></msqrt></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cent</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>c</mi><mo>/</mo><mn>2</mn><mo>,</mo><mi>ap</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>c</mi><mo>,</mo><mn>0</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mi>poligon_regular</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>)</mo><mo>;</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tri</mi><mo>=</mo><mi>triangle</mi><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>cent</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>c</mi><mo>/</mo><mn>2</mn><mo>,</mo><mn>0</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>seg</mi><mo>=</mo><mi>segment</mi><mo>(</mo><mi>cent</mi><mo>,</mo><mi>M</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ang</mi><mo>=</mo><mi>arc</mi><mo>(</mo><mi>cent</mi><mo>,</mo><mi>A</mi><mo>,</mo><mi>M</mi><mo>,</mo><mi>r</mi><mo>/</mo><mn>4</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>=</mo><mi>tauler</mi><mo>(</mo><mi>cent</mi><mo>,</mo><mn>2</mn><mo>*</mo><mi>r</mi><mo>+</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>*</mo><mi>r</mi><mo>+</mo><mn>2</mn><mo>,</mo><mo>{</mo><mi>mostrar_malla</mi><mo>=</mo><mi>fals</mi><mo>,</mo><mi>mostrar_eixos</mi><mo>=</mo><mi>fals</mi><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mi>dibuixa</mi><mo>(</mo><mi>T</mi><mo>,</mo><mo>{</mo><mi>P</mi><mo>,</mo><mi>tri</mi><mo>,</mo><mi>seg</mi><mo>,</mo><mi>ang</mi><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tolerancia</mi><mo>(</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>3</mn></mrow></msup><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>precisio</mi><mo>(</mo><mn>4</mn><mo>)</mo></math></input></command></group></library></session>]]></wirisCasSession><correctAnswers><correctAnswer>#r</correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> Divideix el polígon en triangles isòsceles, i cada triangle en dos triangles rectangles, traçant la seva altura.

Ara ja pots calcular el radi en funció de la meitat del costat; els angles es dedueixen del fet que l'angle desigual del triangle isòsceles és 360º / #n

]]> 1MA.02.1.2.22Q CostatPolígonRegular(Radi) Quant mesura el costat d'un polígon regular de «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»n«/mi»«/mrow»«/mstyle»«/math» costats, si el radi mesura «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»r«/mi»«/mrow»«/mstyle»«/math» cm?
Format: als mil·lèsims i sense unitats.

]]> 1.0000000 0.5000000 0 0 #c <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>5</mn><mo>,</mo><mn>12</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>2</mn><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>4</mn><mo>,</mo><mn>6</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>180</mn><mo>º</mo><mo>/</mo><mi>n</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mfrac><mi>c</mi><mrow><mn>2</mn><mo>*</mo><mi>sin</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow></mfrac><mo>*</mo><mn>1</mn><mo>.</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ap</mi><mo>=</mo><msqrt><mrow><msup><mi>r</mi><mn>2</mn></msup><mo>-</mo><mo>(</mo><mi>c</mi><mo>/</mo><mn>2</mn><msup><mo>)</mo><mn>2</mn></msup></mrow></msqrt></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cent</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>c</mi><mo>/</mo><mn>2</mn><mo>,</mo><mi>ap</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>c</mi><mo>,</mo><mn>0</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mi>poligon_regular</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>)</mo><mo>;</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tri</mi><mo>=</mo><mi>triangle</mi><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>cent</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>c</mi><mo>/</mo><mn>2</mn><mo>,</mo><mn>0</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>seg</mi><mo>=</mo><mi>segment</mi><mo>(</mo><mi>cent</mi><mo>,</mo><mi>M</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ang</mi><mo>=</mo><mi>arc</mi><mo>(</mo><mi>cent</mi><mo>,</mo><mi>A</mi><mo>,</mo><mi>M</mi><mo>,</mo><mi>r</mi><mo>/</mo><mn>4</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>=</mo><mi>tauler</mi><mo>(</mo><mi>cent</mi><mo>,</mo><mn>2</mn><mo>*</mo><mi>r</mi><mo>+</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>*</mo><mi>r</mi><mo>+</mo><mn>2</mn><mo>,</mo><mo>{</mo><mi>mostrar_malla</mi><mo>=</mo><mi>fals</mi><mo>,</mo><mi>mostrar_eixos</mi><mo>=</mo><mi>fals</mi><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mi>dibuixa</mi><mo>(</mo><mi>T</mi><mo>,</mo><mo>{</mo><mi>P</mi><mo>,</mo><mi>tri</mi><mo>,</mo><mi>seg</mi><mo>,</mo><mi>ang</mi><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tolerancia</mi><mo>(</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>3</mn></mrow></msup><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>precisio</mi><mo>(</mo><mn>4</mn><mo>)</mo></math></input></command></group></library></session>]]></wirisCasSession><correctAnswers><correctAnswer>#c</correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">inlineEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> Es divideix el polígon en #n triangles isòsceles.

Es divideix cada triangle isòsceles en dos triangles rectangles, traçant l'altura.

Es resol el triangle rectangle per trobar mig costat del polígon tenint present que l'angle desigual del triangle isòsceles és 360/#n

]]> 1MA.02.1.2.23Q ApotemaPolígonRegular(Radi) Quant mesura l'apotema d'un polígon regular de «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»n«/mi»«/mrow»«/mstyle»«/math» costats, si el radi de la circumferència on està inscrit el polígon és «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»r«/mi»«/math»

Format: als mil·lèsims i sense unitat.

]]> 1.0000000 0.5000000 0 0 #sol <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>5</mn><mo>,</mo><mn>12</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>2</mn><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>4</mn><mo>,</mo><mn>6</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>180</mn><mo>º</mo><mo>/</mo><mi>n</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mfrac><mi>c</mi><mrow><mn>2</mn><mo>*</mo><mi>sin</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow></mfrac><mo>*</mo><mn>1</mn><mo>.</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi><mo>=</mo><msqrt><mrow><msup><mi>r</mi><mn>2</mn></msup><mo>-</mo><mo>(</mo><mi>c</mi><mo>/</mo><mn>2</mn><msup><mo>)</mo><mn>2</mn></msup></mrow></msqrt></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cent</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>c</mi><mo>/</mo><mn>2</mn><mo>,</mo><mi>ap</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>c</mi><mo>,</mo><mn>0</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mi>poligon_regular</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>)</mo><mo>;</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tri</mi><mo>=</mo><mi>triangle</mi><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>cent</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>c</mi><mo>/</mo><mn>2</mn><mo>,</mo><mn>0</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>seg</mi><mo>=</mo><mi>segment</mi><mo>(</mo><mi>cent</mi><mo>,</mo><mi>M</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ang</mi><mo>=</mo><mi>arc</mi><mo>(</mo><mi>cent</mi><mo>,</mo><mi>A</mi><mo>,</mo><mi>M</mi><mo>,</mo><mi>r</mi><mo>/</mo><mn>4</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>=</mo><mi>tauler</mi><mo>(</mo><mi>cent</mi><mo>,</mo><mn>2</mn><mo>*</mo><mi>r</mi><mo>+</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>*</mo><mi>r</mi><mo>+</mo><mn>2</mn><mo>,</mo><mo>{</mo><mi>mostrar_malla</mi><mo>=</mo><mi>fals</mi><mo>,</mo><mi>mostrar_eixos</mi><mo>=</mo><mi>fals</mi><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mi>dibuixa</mi><mo>(</mo><mi>T</mi><mo>,</mo><mo>{</mo><mi>P</mi><mo>,</mo><mi>tri</mi><mo>,</mo><mi>seg</mi><mo>,</mo><mi>ang</mi><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tolerancia</mi><mo>(</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>3</mn></mrow></msup><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>precisio</mi><mo>(</mo><mn>4</mn><mo>)</mo></math></input></command></group></library></session>]]></wirisCasSession><correctAnswers><correctAnswer>#sol</correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">inlineEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> Primer pots calcular el radi del cercle on està inscrit el polígon.

Amb el radi, el costat i el teorema de Pitàgores, pots deduir l'apotema.

]]> 1MA.02.1.2.24 ApotemaPoligonRegular(costat) Quant mesura l'apotema d'un polígon regular de «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»n«/mi»«/mrow»«/mstyle»«/math» costats, si el costat mesura #c cm?

Format: als mil·lèsims i sense unitats

]]> 1.0000000 0.5000000 0 0 #sol <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>5</mn><mo>,</mo><mn>12</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>2</mn><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>4</mn><mo>,</mo><mn>6</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>180</mn><mo>º</mo><mo>/</mo><mi>n</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mfrac><mi>c</mi><mrow><mn>2</mn><mo>*</mo><mi>sin</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow></mfrac><mo>*</mo><mn>1</mn><mo>.</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi><mo>=</mo><msqrt><mrow><msup><mi>r</mi><mn>2</mn></msup><mo>-</mo><mo>(</mo><mi>c</mi><mo>/</mo><mn>2</mn><msup><mo>)</mo><mn>2</mn></msup></mrow></msqrt></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cent</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>c</mi><mo>/</mo><mn>2</mn><mo>,</mo><mi>ap</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>c</mi><mo>,</mo><mn>0</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mi>poligon_regular</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>)</mo><mo>;</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tri</mi><mo>=</mo><mi>triangle</mi><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>cent</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>c</mi><mo>/</mo><mn>2</mn><mo>,</mo><mn>0</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>seg</mi><mo>=</mo><mi>segment</mi><mo>(</mo><mi>cent</mi><mo>,</mo><mi>M</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ang</mi><mo>=</mo><mi>arc</mi><mo>(</mo><mi>cent</mi><mo>,</mo><mi>A</mi><mo>,</mo><mi>M</mi><mo>,</mo><mi>r</mi><mo>/</mo><mn>4</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>=</mo><mi>tauler</mi><mo>(</mo><mi>cent</mi><mo>,</mo><mn>2</mn><mo>*</mo><mi>r</mi><mo>+</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>*</mo><mi>r</mi><mo>+</mo><mn>2</mn><mo>,</mo><mo>{</mo><mi>mostrar_malla</mi><mo>=</mo><mi>fals</mi><mo>,</mo><mi>mostrar_eixos</mi><mo>=</mo><mi>fals</mi><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mi>dibuixa</mi><mo>(</mo><mi>T</mi><mo>,</mo><mo>{</mo><mi>P</mi><mo>,</mo><mi>tri</mi><mo>,</mo><mi>seg</mi><mo>,</mo><mi>ang</mi><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tolerancia</mi><mo>(</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>3</mn></mrow></msup><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>precisio</mi><mo>(</mo><mn>4</mn><mo>)</mo></math></input></command></group></library></session>]]></wirisCasSession><correctAnswers><correctAnswer>#sol</correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> Primer pots calcular el radi del cercle on està inscrit el polígon.

Amb el radi, el costat i el teorema de Pitàgores, pots deduir l'apotema.

]]> 1MA.02.1.2.71Q Ombra edifici(angle?) Un edifici de «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»b«/mi»«/mrow»«/mstyle»«/math» m projecta un ombra de «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»c«/mi»«/mrow»«/mstyle»«/math» m. Quin angle formen els raigs de sol amb el terra en aquell moment?

Format de la resposta: angle en graus, arrodonit i sense unitats

]]>

]]> 1MA 02. RESOLUCIÓ DE TRIANGLES/1MA.02.1 Resolució de triangles rectangles/1MA.02.1.3 Triangles no rectangles (tg) 1MA.02.1.3.11Q ResolAmbTgCimTuró Des del fons d'una vall ombrívola, veiem el cim d'un turó sota un angle de #B1 º. Si ens acostem #d m al turó, l'angle és de #B2 º. Quina és l'altura del turó?

Format de la resposta: arrodonida al centèsims

]]> 1.0000000 0.5000000 0 0 #sol <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>20</mn><mo>,</mo><mn>40</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>10</mn><mo>,</mo><mn>40</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B2</mi><mo>=</mo><mi>B1</mi><mo>+</mo><mi>M</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>10</mn><mo>,</mo><mn>80</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r1</mi><mo>=</mo><mfrac><mrow><mi>B1</mi><mo>*</mo><pi/></mrow><mn>180</mn></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r2</mi><mo>=</mo><mfrac><mrow><mi>B2</mi><mo>*</mo><pi/></mrow><mn>180</mn></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfenced close="|" open="|"><mfrac><mrow><mo>(</mo><mi>d</mi><mo>)</mo><mo>*</mo><mi>tan</mi><mo>(</mo><mi>r1</mi><mo>)</mo><mo>*</mo><mi>tan</mi><mo>(</mo><mi>r2</mi><mo>)</mo></mrow><mrow><mi>tan</mi><mo>(</mo><mi>r1</mi><mo>)</mo><mo>-</mo><mi>tan</mi><mo>(</mo><mi>r2</mi><mo>)</mo></mrow></mfrac></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi><mo>=</mo><mfrac><mrow><mi>arrodoneix</mi><mo>(</mo><mn>100</mn><mo>*</mo><mi>y</mi><mo>)</mo></mrow><mn>100</mn></mfrac><mo>*</mo><mn>1</mn><mo>.</mo><mn>0</mn></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>23</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>23</mn><mo>*</mo><pi/></mrow><mn>180</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>38</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>19</mn><mo>*</mo><pi/></mrow><mn>90</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>77</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>71.57</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer>#sol</correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="precision">8</option><option name="tolerance">10^(-4)</option><option name="relative_tolerance">false</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">textField</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question>

En aquest cas, d = #d.

Calculem les tangents de B1 i B2:

tg B1 = y/(x+#d)

tg B2 = y/x

Aïllem y en les dues equacions i igualem per calcular x.

Un cop calculat x en podem deduir y.

]]> 1MA.02.1.3.12Q ResolTGAvio Dues persones es troben a un distància l'una de l'altra de #d m i observen el pas d'un avió. L'una el veu sota un angle de #B1º i l'altra sota un angle de #B2º. A quina alçada vola l'avió?

Format de la resposta: en metres arrodonida als centèsims

]]> 1.0000000 0.5000000 0 0 #sol <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>20</mn><mo>,</mo><mn>40</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>10</mn><mo>,</mo><mn>40</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B2</mi><mo>=</mo><mi>B1</mi><mo>+</mo><mi>M</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>10</mn><mo>,</mo><mn>80</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r1</mi><mo>=</mo><mfrac><mrow><mi>B1</mi><mo>*</mo><pi/></mrow><mn>180</mn></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r2</mi><mo>=</mo><mfrac><mrow><mi>B2</mi><mo>*</mo><pi/></mrow><mn>180</mn></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfenced close="|" open="|"><mfrac><mrow><mo>(</mo><mi>d</mi><mo>)</mo><mo>*</mo><mi>tan</mi><mo>(</mo><mi>r1</mi><mo>)</mo><mo>*</mo><mi>tan</mi><mo>(</mo><mi>r2</mi><mo>)</mo></mrow><mrow><mi>tan</mi><mo>(</mo><mi>r1</mi><mo>)</mo><mo>-</mo><mi>tan</mi><mo>(</mo><mi>r2</mi><mo>)</mo></mrow></mfrac></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi><mo>=</mo><mfrac><mrow><mi>arrodoneix</mi><mo>(</mo><mn>100</mn><mo>*</mo><mi>y</mi><mo>)</mo></mrow><mn>100</mn></mfrac><mo>*</mo><mn>1</mn><mo>.</mo><mn>0</mn></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>23</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>23</mn><mo>*</mo><pi/></mrow><mn>180</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>38</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>19</mn><mo>*</mo><pi/></mrow><mn>90</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>77</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>71.57</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer>#sol</correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="precision">8</option><option name="tolerance">10^(-4)</option><option name="relative_tolerance">false</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">textField</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question>

En aquest cas, d = #d.

Calculem les tangents de B1 i B2:

tg B1 = y/(x+#d)

tg B2 = y/x

Aïllem y en les dues equacions i igualem per calcular x.

Un cop calculat x en podem deduir y.

]]> 1MA.02.1.3.13Q ResolAmbTg AmpleRiu Sabent que la distància entre dos punts A i B és de «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»m«/mi»«/mrow»«/mstyle»«/math» i que des del punt A es veu un arbre de l'altre costat del riu sota un angle de «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»A«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#186;«/mo»«/mrow»«/mstyle»«/math» i des del punt B sota un angle de «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»B«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#186;«/mo»«/mrow»«/mstyle»«/math». Quina és l'amplada del riu si els dos punts es situen més avall de l'arbre en el curs del riu?

Format: als mil·lèsims i sense unitats.

Calcula les tangents dels dos angles en funció del costat oposat y i dels costats adjacents, x i (#m - x).

Aïlla y i iguala les dues expressions per trobar x.

Substitueix per trobar y.

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1MA.02.1.3.14Q ResolTGArbre+2angles Situats a la riba d’un riu, veiem un arbre sobre l’altra riba sota un angle de «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»A«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#186;«/mo»«/mrow»«/mstyle»«/math». Ens allunyem «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»d«/mi»«/mrow»«/mstyle»«/math» metres cap enrere i aleshores l’angle visual és de «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»B«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#186;«/mo»«/mrow»«/mstyle»«/math». Quina serà l’amplada del riu? i l'altura de l'arbre?

Format: als mil·lèsims i sense unitats

]]> 1.0000000 0.5000000 0 0 amplada=#sol1altura=#sol2]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>A</mi><mo>:</mo><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>30</mn><mo>,</mo><mn>45</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>B</mi><mo>:</mo><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>15</mn><mo>,</mo><mn>30</mn><mo>)</mo></mtd></mtr></mtable><mrow><mi>A</mi><mo>-</mo><mi>B</mi><mo>⩾</mo><mn>5</mn></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>B</mi><mo>⩽</mo><mn>22</mn></mrow><mrow><mi>d</mi><mo>:</mo><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>8</mn><mo>,</mo><mn>12</mn><mo>)</mo></mrow><mrow><mi>d</mi><mo>:</mo><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>7</mn><mo>)</mo></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi><mo>=</mo><mfrac><mrow><mi>d</mi><mo>*</mo><mi>tan</mi><mo>(</mo><mfrac><mrow><mi>B</mi><mo>*</mo><pi/></mrow><mn>180</mn></mfrac><mo>)</mo></mrow><mrow><mi>tan</mi><mo>(</mo><mfrac><mrow><mi>A</mi><mo>*</mo><pi/></mrow><mn>180</mn></mfrac><mo>)</mo><mo>-</mo><mi>tan</mi><mo>(</mo><mfrac><mrow><mi>B</mi><mo>*</mo><pi/></mrow><mn>180</mn></mfrac><mo>)</mo></mrow></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi><mo>=</mo><mi>d</mi><mo>*</mo><mi>tan</mi><mo>(</mo><mfrac><mrow><mi>A</mi><mo>*</mo><pi/></mrow><mn>180</mn></mfrac><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tolerancia</mi><mo>(</mo><mn>0</mn><mo>.</mo><mn>001</mn><mo>)</mo></math></input></command></group></library></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>m</mi><mi>p</mi><mi>l</mi><mi>a</mi><mi mathvariant="normal">d</mi><mi>a</mi><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>1</mn><mspace linebreak="newline"/><mi>a</mi><mi>l</mi><mi>t</mi><mi>u</mi><mi>r</mi><mi>a</mi><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>2</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="precision">8</option><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputCompound">true</data><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> Escriu les tangents dels dos angles en funció de l'altura de l'arbre y i de les dues distàncies x i (x + #d)

]]> 1MA.02.1.3.21Q ResolTGAngleADistànciaTriple

Si ens situem a una distància x d'un arbre, en veiem la part més alta sota un angle de «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»C«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#186;«/mo»«/mrow»«/mstyle»«/math». Sota quin angle el veuríem si ens situéssim a una distància triple?D

Format: en graus, arrodonits i sense unitats.

]]> 1.0000000 0.5000000 0 0 #B <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>:</mo><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>45</mn><mo>,</mo><mn>60</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B1</mi><mo>=</mo><mi>atan</mi><mo>(</mo><mfrac><mrow><mi>tan</mi><mo>(</mo><mfrac><mrow><mi>C</mi><mo>*</mo><pi/></mrow><mn>180</mn></mfrac><mo>)</mo></mrow><mn>3</mn></mfrac><mo>)</mo><mo>*</mo><mfrac><mn>180</mn><pi/></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>=</mo><mi>arrodoneix</mi><mo>(</mo><mi>B1</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tolerancia</mi><mo>(</mo><mn>0</mn><mo>.</mo><mn>001</mn><mo>)</mo></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>49</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20.98</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>21</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer>#B</correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">inlineEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> Escriu les tangents dels angles en funció de l'alçada de l'arbre x i de les dues distàncies x i 3x.

]]> 1MA.02.1.3.31Q altura avió Sabent que la distància entre els punts A i B és de #k,#m km i que des del punt A es veu l'avió amb un amb angle «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»§#945;«/mi»«/math» de #Aº i des del punt B amb un angle «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»§#946;«/mi»«/math» de #Bº. A quina altura vola l'avió?

(Expresseu el resultat en metres amb 3 decimals de precisió)

]]> 1.0000000 0.1000000 0 0 #sol <question><wirisCasSession><session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>:</mo><mo>=</mo><mo> </mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>:</mo><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>100</mn><mo>,</mo><mn>900</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><mi>k</mi><mo>+</mo><mi>m</mi><mo>/</mo><mn>1000</mn></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>:</mo><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>30</mn><mo>,</mo><mn>45</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>:</mo><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>45</mn><mo>,</mo><mn>60</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi><mo>:</mo><mo>=</mo><mfrac><mrow><mi>d</mi><mo>*</mo><mi>tan</mi><mo>(</mo><mi>A</mi><mo>*</mo><mfrac><pi/><mn>180</mn></mfrac><mo>)</mo><mo>*</mo><mi>tan</mi><mo>(</mo><mi>B</mi><mo>*</mo><mfrac><pi/><mn>180</mn></mfrac><mo>)</mo></mrow><mrow><mi>tan</mi><mo>(</mo><mi>A</mi><mo>*</mo><mfrac><pi/><mn>180</mn></mfrac><mo>)</mo><mo>+</mo><mi>tan</mi><mo>(</mo><mi>B</mi><mo>*</mo><mfrac><pi/><mn>180</mn></mfrac><mo>)</mo></mrow></mfrac><mo>*</mo><mn>1000</mn></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tolerància</mi><mo>(</mo><mn>0</mn><mo>.</mo><mn>001</mn><mo>)</mo></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>643</mn><mn>500</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>537.21</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><correctAnswers><correctAnswer type="mathml"> #sol </correctAnswer></correctAnswers><localData><data name="cas">false</data><data name="inputField">textField</data></localData></question> 1MA.02.1.3.32Q amplada d'un riu Sabent que la distància entre els punts A i B és de #m m i que des del punt A es veu un punt de l'altre costat del riu amb un amb angle «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»§#945;«/mi»«/math» de #Aº i des del punt B amb un angle «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»§#946;«/mi»«/math» de #Bº. Quina és l'amplada del riu?
(Expresseu el resultat en metres amb 3 decimals de precisió)

El riu fa {#1} m d'amplada.

]]> 1.0000000 0.1000000 0 <question><wirisCasSession><session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>:</mo><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>100</mn><mo>,</mo><mn>900</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>:</mo><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>30</mn><mo>,</mo><mn>45</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>:</mo><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>45</mn><mo>,</mo><mn>60</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi><mo>:</mo><mo>=</mo><mfrac><mrow><mi>m</mi><mo>*</mo><mi>tan</mi><mo>(</mo><mi>B</mi><mo>º</mo><mo>)</mo></mrow><mrow><mi>tan</mi><mo>(</mo><mi>A</mi><mo>º</mo><mo>)</mo><mo>+</mo><mi>tan</mi><mo>(</mo><mi>B</mi><mo>º</mo><mo>)</mo></mrow></mfrac><mo>*</mo><mi>tan</mi><mo>(</mo><mi>A</mi><mo>º</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tolerància</mi><mo>(</mo><mn>0</mn><mo>.</mo><mn>001</mn><mo>)</mo></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>sol</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn><mo>,</mo><mn>55</mn><mo>,</mo><mn>693</mn><mo>,</mo><mn>284.92</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><localData><data name="cas">false</data></localData></question> 1MA.02.1.3.33Q Arbre distancia triple

Des d'un punt del terra situat a una certa distància a del peu d'un arbre, es veu la part més alta de l'arbre amb un angle «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»§#946;«/mi»«/math» de #Cº. Sota quin angle es veurà si ens col·loquem a una distància que és el triple que l'anterior?
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»§#945;«/mi»«/math» = {#1}º.
(La resposta s'ha de donar en graus sexagesimals amb tres xifres decimals)

]]> 1.0000000 0.1000000 0 <question><wirisCasSession><session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>:</mo><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>45</mn><mo>,</mo><mn>60</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>=</mo><mi>atan</mi><mo>(</mo><mfrac><mrow><mi>tan</mi><mo>(</mo><mfrac><mrow><mi>C</mi><mo>*</mo><pi/></mrow><mn>180</mn></mfrac><mo>)</mo></mrow><mn>3</mn></mfrac><mo>)</mo><mo>*</mo><mfrac><mn>180</mn><pi/></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tolerancia</mi><mo>(</mo><mn>0</mn><mo>.</mo><mn>001</mn><mo>)</mo></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>48</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20.315</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><localData><data name="cas">false</data></localData></question> 1MA.02.1.3.34Q Arbre i 2 angles Situats a la riba d’un riu, veiem un arbre a l’altra riba sota un angle «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»§#945;«/mi»«/math» de #Aº. Ens allunyem #d metres cap enrere i aleshores l’angle visual és «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»§#946;«/mi»«/math» de #Bº. Quina serà l’amplada del riu? {#1} m i l'altura de l'arbre? {#2} m.

(La resposta s'ha de donar en metres amb 3 xifres decimals de precisió)

]]> 2.0000000 0.1000000 0 <question><wirisCasSession><session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>A</mi><mo>:</mo><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>30</mn><mo>,</mo><mn>45</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>B</mi><mo>:</mo><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>15</mn><mo>,</mo><mn>30</mn><mo>)</mo></mtd></mtr></mtable><mrow><mi>A</mi><mo>-</mo><mi>B</mi><mo>&ges;</mo><mn>5</mn></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>B</mi><mo>&les;</mo><mn>22</mn></mrow><mrow><mi>d</mi><mo>:</mo><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>8</mn><mo>,</mo><mn>12</mn><mo>)</mo></mrow><mrow><mi>d</mi><mo>:</mo><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>7</mn><mo>)</mo></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mrow><mi>d</mi><mo>*</mo><mi>tan</mi><mo>(</mo><mfrac><mrow><mi>B</mi><mo>*</mo><pi/></mrow><mn>180</mn></mfrac><mo>)</mo></mrow><mrow><mi>tan</mi><mo>(</mo><mfrac><mrow><mi>A</mi><mo>*</mo><pi/></mrow><mn>180</mn></mfrac><mo>)</mo><mo>-</mo><mi>tan</mi><mo>(</mo><mfrac><mrow><mi>B</mi><mo>*</mo><pi/></mrow><mn>180</mn></mfrac><mo>)</mo></mrow></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi><mo>=</mo><mi>a</mi><mo>*</mo><mi>tan</mi><mo>(</mo><mfrac><mrow><mi>A</mi><mo>*</mo><pi/></mrow><mn>180</mn></mfrac><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tolerancia</mi><mo>(</mo><mn>0</mn><mo>.</mo><mn>001</mn><mo>)</mo></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12.353</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8.3325</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><localData><data name="cas">false</data></localData></question>