Question

Constuct a rectangular box with a square base that holds a given volume V0 cm^3.?
The cost of the material for the sides is 1 cent per cm^2, while the top and bottom costs 3 cents per cm^2. Find, in terms of V0, the dimensions of the box that will minimize the cost of the material. Ratio of these optimal dimensions.

Answer 1

let the base be x by x , and the height be y
x^2 y = VO
y= VO/x^2 , where VO is a constant

cost = 1(xy) + 3x^2
= x(VO/x^2) + 3x^2 = VO/x + 3x^2
d(cost)/dx = - VO/x^2 + 6x = 0 for a min cost

6x = VO/x^2
x^3 = VO/6
x = (VO/6)^(1/3)

I will let you sub that into y = VO/x^2