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Ejercicios de Optimizacion3

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

Summary: Ejercicios de Optimizacion

OPTIMIZACION3

1. Una caja con base cuadrada y parte superior abierta debe tener un volumen de 50cm cubicos. encuentre las dimensiones de la caja que minimicen la cantidad de material que va a ser usado.

V=50

V = x 2 y V = x 2 y
(1)

Area de la caja sin volumen

A = x 2 + 4 xy A = x 2 + 4 xy
(2)
y = 50 x 2 y = 50 x 2
(3)

sustituimos en la del area.

A ( x ) = x 2 + 4 x ( 50 x 2 ) ⇒ x 2 + 200 x ⇒ x 2 + 200 x - 1 A ( x ) = x 2 + 4 x ( 50 x 2 ) ⇒ x 2 + 200 x ⇒ x 2 + 200 x - 1
(4)
A ' ( x ) = 2 x - 200 x - 2 ⇒ 2 x + 200 x 2 ⇒ 2 x 3 - 200 x 2 A ' ( x ) = 2 x - 200 x - 2 ⇒ 2 x + 200 x 2 ⇒ 2 x 3 - 200 x 2
(5)
A ' ' ( x ) = 2 + 200 ( 2 x 3 ) ⇒ 2 + 400 x 3 > 0 A ' ' ( x ) = 2 + 200 ( 2 x 3 ) ⇒ 2 + 400 x 3 > 0
(6)

calculamos los puntos criticos.

A ' ( x ) = 0 ⇒ 2 x 3 - 200 = 0 ⇒ x 3 = 100 ⇒ x = 100 3 cm A ' ( x ) = 0 ⇒ 2 x 3 - 200 = 0 ⇒ x 3 = 100 ⇒ x = 100 3 cm
(7)
y = 50 x 2 ⇒ 50 100 2 3 ⇒ 1 2 100 100 2 3 ⇒ 1 2 100 3 ⇒ 1 2 x cm y = 50 x 2 ⇒ 50 100 2 3 ⇒ 1 2 100 100 2 3 ⇒ 1 2 100 3 ⇒ 1 2 x cm
(8)

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