Posted by Caitlin on .
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,
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,
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,
Yes, you're right.
If we proceed with
Cost = 6 w^2 + 8 w h
we get:
w=6 and h=9.