For a closed for a closed rectangular box, with a square base x by x cm and height h cm, find the dimensions giving the minimum surface area, given that the volume is 7 cm3 box, with a square base x by x cm and height h cm, find the dimensions giving the minimum surface area, given that the volume is 7 cm3. enclose arguments of functions, numerators, and denominators in parentheses. for example, sin(2x) or (a−b)/(1+n).

  x = h = ∛7 cm ≈ 1.913 cm

Step-by-step explanation:

The volume and other dimensions are related by ...

  V = Bh = x²h

Solving for h gives ...

  h = V/(x²)

The surface area is ...

  S = 2(x² +h(2x)) = 2x² +4x(V/(x²)) = 2x² +4V/x

Differentiating with respect to x, we can find where the derivative is zero.

  S' = 4x -4v/(x²) = 0

  x³ -V = 0 . . . . . . . multiply by x²/4

  x = ∛V . . . . . . . . . solve for x

  h = V/(∛V)² = (∛V)³/(∛V)² = ∛V

The surface area is minimized when the box is a cube. Its edge lengths are all (∛7) cm ≈ 1.913 cm.

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step-by-step explanation: