A rectangular tank with a square​ base, an open​ top, and a volume of 4,000 ft cubed is to be constructed of sheet steel. Find the dimensions of the tank that has the minimum surface area.

Question

I'm not understanding how to get started and find the optimization function (for any of these problems).

Answer 1

side = x
height = y

volume = x^2y = 4000 so y = 4000/x^2
area = A = x^2 + 4 x y

A = x^2 + 4 x (4000/x^2)
A = x^2 + 16000/x
min or max when dA/dx = 0
0 = 2x -16000/x^2
16000 = 2 x^3
x^3 = 16000

x = 25.2
y = 4000/x^2 = 6.3