CÁLCULO/2. FUNCIONES FUNCIONES 2.2.5.- Aplicación función exponencial: ley de newton La ley del enfriamiento de Newton establece que la temperatura de un objeto caliente disminuye en forma exponencial con el tiempo hacia la temperatura del ambiente. Es decir, la temperatura «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»u«/mi»«/math» de un objeto caliente en un instante «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«/math» satisface la ecuación

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable»«mtr»«mtd»«mi»u«/mi»«mo»=«/mo»«mi»T«/mi»«mo»+«/mo»«mfenced»«mrow»«msub»«mi»u«/mi»«mn»0«/mn»«/msub»«mo»-«/mo»«mi»T«/mi»«/mrow»«/mfenced»«msup»«mi»e«/mi»«mrow»«mi»k«/mi»«mi»t«/mi»«/mrow»«/msup»«/mtd»«mtd»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«/mtd»«mtd»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mi»k«/mi»«mo»§lt;«/mo»«mn»0«/mn»«/mtd»«/mtr»«/mtable»«/math»

donde «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»T«/mi»«/math» es la temperatura constante del ambiente, «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»u«/mi»«mn»0«/mn»«/msub»«/math» la temperatura inicial del objeto caliente y «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»k«/mi»«/math» un número negativo. Un objeto se calienta a «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»#uini1«/mi»«/math» y después se deja enfriar en un cuarto cuya temperatura del aire es de «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»#Temp1«/mi»«/math». Si después de «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»#tini1«/mi»«/math», la temperatura del objeto es de «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»#ufin1«/mi»«/math».

#preg
]]> Solución:

Usando la ecuación que describe la ley del enfriamiento de Newton

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable»«mtr»«mtd»«mi»#ECUA«/mi»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«mtd»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mfenced»«mn»1«/mn»«/mfenced»«/mtd»«/mtr»«/mtable»«/math»

cuando «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»T«/mi»«mo»=«/mo»«mi»#T1«/mi»«/math» y «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»u«/mi»«mn»0«/mn»«/msub»«mo»=«/mo»«mi»#uini1«/mi»«/math», se tiene que la temperatura «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»u«/mi»«/math», en grados Celsius, del objeto en el instante «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«/math», en minutos, es

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»#ecua1«/mi»«/math»

donde «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»k«/mi»«/math» es un número negativo.
Determinaremos el valor de la constante «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»k«/mi»«/math».
Como el problema dice que a los «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«mo»=«/mo»«mi»#tini1«/mi»«/math» la temperatura es «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»u«/mi»«mo»=«/mo»«mi»#ufin1«/mi»«/math», reemplazamos dichos valores en la ecuación «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfenced»«mn»1«/mn»«/mfenced»«/math», obteniendo

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable»«mtr»«mtd»«mi»#uf1«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»#ecua2«/mi»«/mtd»«mtd»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»/«/mo»«mo»+«/mo»«mfenced»«mrow»«mo»-«/mo»«mi»#TA1«/mi»«/mrow»«/mfenced»«/mtd»«/mtr»«mtr»«mtd»«mi»#ecua3«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»#ecua4«/mi»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«/mtr»«mtr»«mtd»«mi»#uf3«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»#ecua4«/mi»«/mtd»«mtd»«mo»/«/mo»«mo»·«/mo»«mfenced»«mfrac»«mn»1«/mn»«mrow»«mi»#a1«/mi»«/mrow»«/mfrac»«/mfenced»«/mtd»«/mtr»«mtr»«mtd»«mi»#uf4«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»#ecua5«/mi»«/mtd»«mtd»«mo»/«/mo»«mi mathvariant=¨normal¨»ln«/mi»«mfenced»«mo»§nbsp;«/mo»«/mfenced»«/mtd»«/mtr»«mtr»«mtd»«mi mathvariant=¨normal¨»ln«/mi»«mfenced»«mrow»«mi»#uf4«/mi»«/mrow»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi mathvariant=¨normal¨»ln«/mi»«mfenced»«mrow»«mi»#ecua5«/mi»«/mrow»«/mfenced»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«/mtr»«mtr»«mtd»«mi»#uf5«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»#ecua6«/mi»«/mtd»«mtd»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»/«/mo»«mo»·«/mo»«mfenced»«mfrac»«mn»1«/mn»«mrow»«mi»#ti1«/mi»«/mrow»«/mfrac»«/mfenced»«/mtd»«/mtr»«mtr»«mtd»«mfrac»«mrow»«mi»#uf5«/mi»«/mrow»«mrow»«mi»#ti1«/mi»«/mrow»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»k«/mi»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«/mtr»«mtr»«mtd»«mi»#uf7«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»k«/mi»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«/mtr»«/mtable»«/math»

Reemplazando el valor de «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»k«/mi»«/math» en la ecuación «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfenced»«mn»2«/mn»«/mfenced»«/math» se tiene

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable»«mtr»«mtd»«mi»#ecua7«/mi»«/mtd»«mtd»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»(«/mo»«mn»3«/mn»«mo»)«/mo»«/mtd»«/mtr»«/mtable»«/math»

#ret1

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable»«mtr»«mtd»«mi»#up11«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»#ecua8«/mi»«/mtd»«/mtr»«mtr»«mtd»«mi»#uf8«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»#ecua9«/mi»«/mtd»«/mtr»«mtr»«mtd»«mi»#uf9«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»#ecua10«/mi»«/mtd»«/mtr»«mtr»«mtd»«mi»#uf10«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»#ecua11«/mi»«/mtd»«/mtr»«mtr»«mtd»«mi»#uf11«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»#ecua12«/mi»«/mtd»«/mtr»«mtr»«mtd»«mi»#ecua13«/mi»«/mtd»«mtd»«mi»#uf13«/mi»«/mtd»«mtd»«mi»#uf12«/mi»«/mtd»«/mtr»«mtr»«mtd»«mo»§nbsp;«/mo»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«/mtr»«mtr»«mtd»«mo»§nbsp;«/mo»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«mtd»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«/mtd»«/mtr»«/mtable»«/math»

#ret2

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definitionURL="http://.../units/degree/angular">°</csymbol><csymbol definitionURL="http://.../units/coulomb">C</csymbol><mo>&quot;</mo><mo>,</mo><mi>uf1</mi></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tini1</mi><mo>=</mo><mi>sustituir_cadena</mi><mfenced><mrow><mo>&quot;</mo><mo>#</mo><mn>1</mn><mo> </mo><mi>minutos</mi><mo>&quot;</mo><mo>,</mo><mi>ti1</mi></mrow></mfenced></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"><apply><csymbol 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xmlns="http://www.w3.org/1998/Math/MathML"><mi>ecua4</mi><mo>=</mo><mi>sustituir_cadena</mi><mfenced><mrow><mo>&quot;</mo><mo>#</mo><mn>1</mn><msup><exponentiale/><mrow><mo>#</mo><mn>3</mn><mo>#</mo><mn>2</mn></mrow></msup><mo>&quot;</mo><mo>,</mo><mi>a1</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>ti1</mi></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ecua5</mi><mo>=</mo><mi>sustituir_cadena</mi><mfenced><mrow><mo>&quot;</mo><msup><exponentiale/><mrow><mo>#</mo><mn>2</mn><mo>#</mo><mn>1</mn></mrow></msup><mo>&quot;</mo><mo>,</mo><mi>k</mi><mo>,</mo><mi>ti1</mi></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>uf4</mi><mo>=</mo><mfrac><mi>uf3</mi><mi>a1</mi></mfrac></math></input></command><command><input><math 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open="&verbar;"><mrow><mi>x</mi><mo>-</mo><mi>sol</mi></mrow></mfenced><mo>&ges;</mo><mn>0</mn><mo>.</mo><mn>1</mn></mrow></mfenced><mo>?</mo></mtd></mtr></mtable></apply></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>uf12</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3.5736</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><assertions><assertion name="syntax_expression" correctAnswer="0"/><assertion name="equivalent_function" correctAnswer="0"><param name="name">test1</param></assertion></assertions><localData><data name="inputField">textField</data><data name="inputCompound">false</data><data name="cas">false</data></localData></question>