Question

A rectangular box with a square bottom and a volume of 256 cubic feet is to be constructed. The top and bottom cost $ .10 per square foot to make and the four sides cost $ .05 per square foot to make. Find the approximate dimensions of the box which would minimize its cost.

Answer 1

let each side of the base be x ft
let the height of the box be y ft
V= (x^2)(y) = 256
y = 256/x^2

form the cost equation ....
C = 2(.10)x^2 + 4(.05)xy
= .2x^2 + 2x(256/x^2)
= .2x^2 + 512/x

C' = .4x - 512/x^2 = 0 for min of C

I get x^3 = 1280
x = 10.86 and y = 2.17

check my arithmetic