Question

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 18 cm3.

Answer 1

∛18 * ∛18 * 18/(∛18)²

Explanation:

Let the surface area of the box be expressed as S = 2(LB+BH+LH) where

L is the length of the box = x

B is the breadth of the box = x

H is the height of the box = h

Substituting this variables into the formula, we will have;

S = 2(x(x)+xh+xh)

S = 2x²+2xh+2xh

S = 2x² + 4xh and the Volume V = x²h

If V = x²h; h = V/x²

Substituting h = V/x² into the surface area will give;

S = 2x² + 4x(V/x²)

Since the volume V = 18cm³

S = 2x² + 4x(18/x²)

S = 2x² + 72/x

Differentiating the function with respect to x to get the minimal point, we will have;

dS/dx = 4x - 72/x²

at dS/dx = 0

4x - 72/x² = 0

- 72/x² = -4x

72 = 4x³

x³ = 72/4

x³ = 18

`x = \sqrt[3]{18}`

Critical point is at
`x = \sqrt[3]{18}`

If x²h = 18

(∛18)²h =18

h = 18/(∛18)²

Hence the dimension is ∛18 * ∛18 * 18/(∛18)²