Question

A closed box is constructed of 4200cm2 of cardboard. The box is a cuboid, with height hcm and square base of side xcm . What is the value of x which maximises the volume of the box? Give your answer in cm correct to 3 significant figures. [Answer format: 00.0 cm]

Answer 1

x is 10
`√(7)`

Explanation:

• Let x is the side of the base
• Let h is the height

Given that:

• The area A = 4200 cm2

<=> 2
`x^(2)` + 4xh = 4200 cm2

<=> 4xh = 4200 - 2
`x^(2)`

<=> h = (4200 - 2
`x^(2)` )/4x

• The volume of the box:

V =
`x^(2)` h

<=> V =
`x^(2)` (4200 - 2
`x^(2)` )/4x

<=> V = (4200x - 2
`x^(3)` )/4

• To find the maximum volume, we differentiate volume with respect to x:

dV/dx = (4200 - 6
`x^(2)` )/4

Set dV/dx = 0, we have:

4200 - 6
`x^(2)` =0

<=>
`x^(2)` = 700

<=> x = 10
`√(7)`

• We differentiate again

d²V/dx² = -12x/4

It is negative so the volume is maximum.

So x is 10
`√(7)`