Butch is making a closed rectangular box with a square base from two different metals. The top and bottom is made from tungsten which costs $6/cm^2, and the sides are made from titanium which costs $32/cm^2.
If the box must have a volume of 800cm^3, determine the dimensions that minimize the overall cost.
let each side of the base be x cm
let the height be y cm
volume = x^2y
x^2 y = 800
y = 800/x^2
cost = 6(top+ bottom) + 32(4 equal sides)
= 6(2x^2) + 128xy
= 12x^2 + 128x(800/x^2)
= 12x^2 + 102400/x
d(cost)/dx = 24x - 102400/x^2 = 0 for a min of cost
24x = 102400/x^2
x^2 = 12800/3
x = 16.22 cm
y = 3.04 cm
check my arithmetic