Question

A rectangular tank with a square​ base, an open​ top, and a volume of 8 comma 7888,788 ft cubedft3 is to be constructed of sheet steel. Find the dimensions of the tank that has the minimum surface area.

Answer 1

Answer:

  26 ft square by 13 ft high

Step-by-step explanation:

The tank will have minimum surface area when opposite sides have the same total area as the square bottom. That is, their height is half their width. This makes the tank half a cube. Said cube would have a volume of ...

  2·(8788 ft^3) = (26 ft)^3

The square bottom of the tank is 26 ft square, and its height is 13 ft.

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Solution using derivatives

If x is the side length of the square bottom, the height is 8788/x^2 and the area is ...

  x^2 + 4x(8788/x^2) = x^2 +35152/x

The derivative of this is zero when area is minimized:

  2x -35152/x^2 = 0

  x^3 = 17576 = 26^3 . . . . . multiply by x^2/2, add 17576

  x = 26

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As the attached graph shows, a graphing calculator can also provide the solution.