Question

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).

Answer 1

x = h = ∛7 cm ≈ 1.913 cm

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.