Posted by **Caitlin** on Tuesday, June 5, 2012 at 8:53pm.

A closed box with a square base is to be constructed so that its volume is 324 cubed feet. The material for the top and bottom cost is $3 per square foot, and that for the sides $2 per square foot. Find the dimensions of the box so that the cost will be minimum.

- Calc -
**Damon**, Tuesday, June 5, 2012 at 9:01pm
top and bottom are w by w

height is h

324 = w^2 h so h = 324/w^2

Cost = 6 w^2 + 4 w h

= 6 w^2 + 4 w (324/w^2)

= 6 w^2 + 1296/w

dC/dw = 0 for min = 12 w -1296/w^2

0 = 12 w^3 - 1296

w^3 = 108

w = 4.76

h = 14.3

- Calc -
**Caitlin**, Tuesday, June 5, 2012 at 9:10pm
Are you sure about the 4w (324/w^2)?

Shouldn't it be 8w (324/w^2), since you are multiplying the four sides by the cost of $2 per square foot?

- Calc -
**MathMate**, Tuesday, June 5, 2012 at 10:21pm
Yes, you're right.

If we proceed with

Cost = 6 w^2 + 8 w h

we get:

w=6 and h=9.

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