Question

A rectangular prism has a volume given by the expression (x^4+ 5x^3– 64x - 320) m^2

If the height of the
prism is (x + 5) m, then determine the possible length and width of the prism.

Answer 1

The possible length and width of the prism are
`(x-4)m` and
`(x^2+4x+16)m`.

Explanation:

The base of the rectangular prism is rectangular. Let
`l` and
`w` be the length and width of the base.

Area of the base
`= lw`

Given that the volume,

`V=(x^4+ 5x^3– 64x - 320) m^2`,

and the height,

`h= (x+5) m`.

The volume of the prism = (Area of the base) x (Height).i.e.

`V=lwh`

`\Rightarrow lw =(V)/(h)`

`\Rightarrow lw =((x^4+ 5x^3– 64x - 320))/((x+5))`

`\Rightarrow lw =((x+5)(x^3-64))/((x+5))`

`\Rightarrow lw =x^3-64`

`\Rightarrow lw =x^3-4^3`

`\Rightarrow lw =(x-4)(x^2+4x+16)` [ using the identity
`p^3-r^3=(p-r)(p^2+pr+r^2)]`

Hence, the possible length and width of the prism are
`(x-4)m` and
`(x^2+4x+16)m`.