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Ejercicio De Optimizacion

Module by: DIEGO DIAZ LEON. E-mail the author

Summary: Ejercicio de Optimizacion

1. Un ranchero tiene 300m de malla para cercar dos corrales rectangulares iguales y contiguos, es decir, que comparten un lado de la cerca. Determine las dimensiones de los corrales para que el area deseada sea maxima.

Se despeja la variable Y de la formula del perimetro ya que es el unico dato que tenemos.

p = 4 x + 3 y = 300 & A = 2 xy p = 4 x + 3 y = 300 & A = 2 xy
(1)
y = 300 - 4 x 3 y = 300 - 4 x 3
(2)
A ( x ) = 2 x ( 300 - 4 x ) 3 = 200 x - 8 3 x 2 A ( x ) = 2 x ( 300 - 4 x ) 3 = 200 x - 8 3 x 2
(3)

Al derivar obtenemos los puntos criticos:

A ' ( x ) = 200 - 16 3 x = 0 ⇒ 16 3 x = 200 ⇒ 3 * 200 16 = 75 2 A ' ( x ) = 200 - 16 3 x = 0 ⇒ 16 3 x = 200 ⇒ 3 * 200 16 = 75 2
(4)

Con la segunda derivada se halla los maximo:

A ' ' ( x ) = - 16 3 < 0 A ' ' ( x ) = - 16 3 < 0
(5)

como es menor que cero es un maximo.

El area maxima ocurre para:

x = 75 2 m & y = 300 - 150 3 = 50 m x = 75 2 m & y = 300 - 150 3 = 50 m
(6)

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