Question

A rectangular box, with a lid, is made from thin metal. Its length = 2x cm and its width = x cm. If the box must have a volume of 72 cm^3,
a) Show that the surface area A cm^2 of metal used is given by A = 4x^2 + 216/x,
b) Find the value of x so that A is a minimum.
c) Find the minimum Area.

Answer 1

The value of x for a minimum surface area is 3 cm producing a surface area of 108 cm³

Volume

The volume of a box is given by:

Volume = length * width * height

72 = 2x * x * height

Height = 36/x²

Surface area (A) = 2(length * height + length * width + width * height)

A = 2(2x * x + 2x * 36/x² + 36/x² * x) = 2(2x² + 72/x + 36/x)

A = 4x² + 216/x

The minimum area is at dA/dx = 0, hence:

dA/dx = 8x - 216/x²

8x - 216/x² = 0

8x = 216/x²

8x³ = 216

x = 3 cm

A = 4(3)² + 216/3 = 108 cm³

The value of x for a minimum surface area is 3 cm producing a surface area of 108 cm³

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