A rectangular box with a volume of 64 ftcubed is to be constructed with a square base and top. The cost per square foot for the bottom is 15cents​, for the top is 10cents​, and for the sides is 2.5cents. What dimensions will minimize the​ cost?

Answer 1

If the base has side x, and the height is y, then
x^2 y = 64
The cost is
c(x,y) = 15x^2 + 10x^2 + 2.5 * 4xy
But, y = 64/x^2, so
c(x) = 25x^2 + 640/x
Now just find where c'(x) = 0