Your surface area consists of a square base and the 4 equal rectangles forming the sides
let the side of the base be x, and the height of the box y
then x^2 y = 32
y = 32/x^2
SA = x^2 + 4xy
= x^2 + 4(32/x^2) = x^2 + 128x^-2
dSA/dx = 2x - 256/x^3 = 0 for a min of SA
2x = 256/x^3
x^4 = 128
x = ± ..... , ignore the negative, find the corresponding y
A rectangular tank with a square base, an open top, and a volume of 32 ft cubed is to be constructed of sheet steel. Find the dimensions of the tank that has the minimum surface area.
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