You have been asked to design a rectangular box with a square base and an open top. The volume of the box must be 2916cm3. Determine the dimensions of the bin that will minimize the surface area, where x is the length of each side of the base and y is the height of the box. Enter an exact answer.

  x = 18 cm

  y = 9 cm

Step-by-step explanation:

An open-top box will have minimum surface area when it is in the shape of half a cube. The whole cube would have volume 2×2916 = 5832 cm³, so its side length would be ∛5832 cm = 18 cm.

The dimensions of the box are x = 18 cm; y = 9 cm.

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The surface area is x^2 +4xy, where y = 2916/x^2. That is, the surface area is ...

  S = x^2 +11664/x

Setting the derivative to zero, we find ...

  dS/dx = 0 = 2x -11664/x^2

  x^3 = 5832 . . . . . . . . may look familiar

  x = ∛5832 = 18

  y = 2916/18^2 = 9