Asked by Abby
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.
Answers
Answered by
Reiny
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
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
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