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Ejercicio de Optimizacion 2

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

Summary: Ejercicio de Optimizacion

2. Un terreno tiene la forma de un rectangulo con dos semicirculos en los extremos. Si el perimetro del terreno es de 50m, encontrar las dimensiones del terreno para que tenga el area maxima.

A = 2 xy + π x 2 A = 2 xy + π x 2
(1)

 

p = 50 m & p = 2 y + 2 π x p = 50 m & p = 2 y + 2 π x
(2)
2 y + 2 π x = 50 ⇒ y = 50 - 2 π x 2 = 25 - π x 2 y + 2 π x = 50 ⇒ y = 50 - 2 π x 2 = 25 - π x
(3)
A ( x ) = 2 x ( 25 - π x ) + π x 2 ⇒ 50 x + x 2 ( π - 2 π ) ⇒ 50 x - π x 2 A ( x ) = 2 x ( 25 - π x ) + π x 2 ⇒ 50 x + x 2 ( π - 2 π ) ⇒ 50 x - π x 2
(4)
A ' ( x ) = ( 50 x - π x 2 ) ⇒ 50 - 2 π x = 0 ⇒ x = 50 2 x = 25 x A ' ( x ) = ( 50 x - π x 2 ) ⇒ 50 - 2 π x = 0 ⇒ x = 50 2 x = 25 x
(5)
A ' ' ( x ) = - 2 π < 0 Es un maximo . A ' ' ( x ) = - 2 π < 0 Es un maximo .
(6)

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