Question

A rectangular box has the dimensions shown in the diagram. The volume of the box is given by the function V(x) = x3 – 4x, where x is the height in inches.

What is the height of the box if the volume is 15 in.3?


There is a rectangular box with length x+2 in. , width x-2 in. , and height x in.


height = in.

Answer 1

`Height =3`

Explanation:

Given

`Length = x + 2`

`Width = x - 2`

`Height = x`

`Volume = 15`

Required

Find the height

The volume is calculated as thus:

`Volume = Length * Width * Height`

This gives

`15 = (x + 2) * (x - 2) * x`

Apply difference of two squares

`15 =(x^2 - 4 ) * x`

`15 = x^3 - 4x`

Subtract 15 from both sides

`x^3 - 4x - 15 = 0`

Using a calculator to factorize, we have:

`(x-3)(x^2+3x+5) = 0`

Split

`x - 3 = 0` or
`x^2 + 3x + 5 = 0`

`x = 3` or
`x^2 + 3x + 5 = 0`

x has complex roots for
`x^2 + 3x + 5 = 0`

Hence:

`x = 3`

Recall that:

`Height = x`

`Height =3`