Question

A company wishes to manufacture a box with a volume of 40 cubic feet that is open on top and is twice as long as it is wide. Find the width of the box that can be produced using the minimum amount of material.

Answer 1

The width of the box is 3 ft that can be produced using the minimum amount of material.

What are the volume and surface area of a box?

Consider a box with length 'l', width 'w' and height 'h',

Volume V = l × w × h

Surface area S = 2(lw + wh + hl)

Calculation:

It is given that,

Volume of the box V = 36 cubic feet and l = 2w

Using the formulae,

V = 2w × w × h = 2w²h ...(1)

S = (2w² + 2wh + 4wh) = 2w² + 6wh ... (2) (here only single lw is considered since it is open top)

From equation (1),

36 = 2w²h

⇒ w²h = 18

⇒ h =18/w²

equation (2) becomes

S = 2w² + 6w(18/w²) = 2w² + 108/w ...(3)

So, for a minimum amount of material, the width of the box is,

On differentiating the surface area w.r.t 'w'

S' = 4w - 108/w²

For a minimum amount, substituting S' = 0 we get,

0 = 4w - 108/w²

⇒ 4w = 108/w²

⇒ 4w³ = 108

⇒ w³ =  27

∴ w = 3 feet

Therefore, the width of the given box is 3 feet.

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