8. Progresiones PROGRESIÓN GEOMÉTRICA (bacterias) Un tipo de bacterias se reproducen por bipartición cada #tiempo . ¿Cuántas bacterias habrá después de #t horas?

]]> Si una bacteria se reproduce por bipartición significa que cada bacteria se parte en dos y por lo tanto se generan 2 bacterias. Considerando que este proceso se repite continuamente se define una Progresión Geométrica (P.G), de primer término 1 ( número de bacterias inicial ) y razón 2.

La expresión que representa la reproducción de estas bacterias está dada por:

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mathvariant=¨bold-italic¨»s«/mi»«mo mathvariant=¨bold¨»§#160;«/mo»«mi mathvariant=¨bold-italic¨»e«/mi»«mi mathvariant=¨bold-italic¨»s«/mi»«mo mathvariant=¨bold¨»:«/mo»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«msub»«mi mathvariant=¨bold-italic¨»a«/mi»«mi mathvariant=¨bold¨»n«/mi»«/msub»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»=«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»§#183;«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«msup»«mn mathvariant=¨bold¨»2«/mn»«mrow»«mi mathvariant=¨bold¨»n«/mi»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/msup»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«mi mathvariant=¨bold-italic¨»q«/mi»«mi mathvariant=¨bold-italic¨»u«/mi»«mi mathvariant=¨bold-italic¨»e«/mi»«mo mathvariant=¨bold¨»§#160;«/mo»«mi mathvariant=¨bold-italic¨»e«/mi»«mi mathvariant=¨bold-italic¨»q«/mi»«mi mathvariant=¨bold-italic¨»u«/mi»«mi mathvariant=¨bold-italic¨»i«/mi»«mi mathvariant=¨bold-italic¨»v«/mi»«mi mathvariant=¨bold-italic¨»a«/mi»«mi mathvariant=¨bold-italic¨»l«/mi»«mi mathvariant=¨bold-italic¨»e«/mi»«mo mathvariant=¨bold¨»§#160;«/mo»«mi mathvariant=¨bold-italic¨»a«/mi»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»:«/mo»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«msub»«mi mathvariant=¨bold-italic¨»a«/mi»«mi mathvariant=¨bold¨»n«/mi»«/msub»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»=«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«msup»«mn mathvariant=¨bold¨»2«/mn»«mrow»«mi mathvariant=¨bold¨»n«/mi»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/msup»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«/math»

Se requiere determinar cuántas bacterias habrá después de #t horas. Si consideramos que la bipartición se produce cada

#tiempo, entonces debemos determinar cuántas #tiempo corresponden #t horas:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»e«/mi»«mi»s«/mi»«mo»§#160;«/mo»«mi»a«/mi»«mi»s«/mi»«mi»i«/mi»«mo»§#160;«/mo»«mi»c«/mi»«mi»o«/mi»«mi»m«/mi»«mi»o«/mi»«mo»§#160;«/mo»«mo»#«/mo»«mi»t«/mi»«mo»§#160;«/mo»«mi»h«/mi»«mi»o«/mi»«mi»r«/mi»«mi»a«/mi»«mi»s«/mi»«mo»§#160;«/mo»«mi»c«/mi»«mi»o«/mi»«mi»r«/mi»«mi»r«/mi»«mi»e«/mi»«mi»s«/mi»«mi»p«/mi»«mi»o«/mi»«mi»n«/mi»«mi»d«/mi»«mi»e«/mi»«mi»n«/mi»«mo»§#160;«/mo»«mi»a«/mi»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»#«/mo»«mi»i«/mi»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»#«/mo»«mi»t«/mi»«mi»i«/mi»«mi»e«/mi»«mi»m«/mi»«mi»p«/mi»«mi»o«/mi»«/math»

Tomando en cuenta que la primera bipartición se obtiene con «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»a«/mi»«mn»2«/mn»«/msub»«/math», debemos calcular «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»a«/mi»«mrow»«mo»#«/mo»«mi»j«/mi»«/mrow»«/msub»«/math» , es decir:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»a«/mi»«mrow»«mo»#«/mo»«mi»j«/mi»«/mrow»«/msub»«mo»=«/mo»«mo»§#160;«/mo»«msup»«mn»2«/mn»«mrow»«mo»#«/mo»«mi»i«/mi»«/mrow»«/msup»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«msub»«mi»a«/mi»«mrow»«mo»#«/mo»«mi»j«/mi»«/mrow»«/msub»«mo»=«/mo»«mo»§#160;«/mo»«mo»#«/mo»«mi»s«/mi»«mi»o«/mi»«mi»l«/mi»«/math»

Luego después de #t horas habrá #sol bacterias

]]> 1 .3333333 0 0 #sol Muy bien

]]> * Al parecer cometiste un error, revisa tu desarrollo y consulta a tu profesor si aún te quedan dudas.

]]> <question><wirisCasSession><![CDATA[<session lang="es" 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>progresion</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><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>aleatorio</mi><mo>(</mo><mn>1</mn><mo>.</mo><mo>.</mo><mn>10</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tiempo</mi><mo>=</mo><mi>aleatorio</mi><mo>[</mo><mo>"</mo><mi>cuarto</mi><mo> </mo><mi>de</mi><mo> </mo><mi>hora</mi><mo>"</mo><mo>,</mo><mo>"</mo><mi>media</mi><mo> </mo><mi>hora</mi><mo>"</mo><mo>,</mo><mo>"</mo><mi>diez</mi><mo> </mo><mi>minutos</mi><mo>"</mo><mo>]</mo></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 definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>tiempo</mi><mo>=</mo><mo>"</mo><mi>cuarto</mi><mo> </mo><mi>de</mi><mo> </mo><mi>hora</mi><mo>"</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>4</mn><mo>*</mo><mi>t</mi></mrow><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>tiempo</mi><mo>=</mo><mo>"</mo><mi>media</mi><mo> </mo><mi>hora</mi><mo>"</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>2</mn><mo>*</mo><mi>t</mi></mrow><mrow><mi>i</mi><mo>=</mo><mn>6</mn><mo>*</mo><mi>t</mi></mrow></apply></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn>1</mn></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi><mo>=</mo><mo> </mo><mi>a</mi><mo>(</mo><mi>j</mi><mo>)</mo></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tiempo</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><ms>media</ms><mo> </mo><ms>hora</ms></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></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>4</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><correctAnswer id="1">*</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>