Need help with setting up optimization problem

Sep 2017
200
2
nj
So, i got the following problem:

A closed box with rectangular sides is built according to the following specifications: The top and bottom sides are made of a material that costs 5 dollars per square foot. the four vertical sides are made of a material that costs 3 dollars per square foot. The top and bottom sides are rectangles with sides of length x and y, where 2x = 7y. The total cost of materials is 100 dollars. Find the largest possible volume that such a box can hold.


My functions are:

Cost= 10xy + 12zy = 100
V = xyz

2x = y7

I'm trying to put Volume in terms of one variable, but i cant... am i missing something?
 

romsek

MHF Helper
Nov 2013
6,828
3,073
California
take the cost function with

$x = \dfrac 7 2 y$

$35 y^2 + 12 z y =100$

$z = \dfrac{100-35y^2}{12y}$

now you have

$V = x y z = \dfrac 7 2 y \cdot y \cdot \dfrac{100-35y^2}{12y} = \dfrac{7y(100-35y^2)}{24}$

and this can be maximized in the usual way.
 
Sep 2017
200
2
nj
Omg thank. Didn't think that way!! Thanks