Question

The volume of a rectangular solid is given by the polynomial 3x^4-3x^3-33x^2+54x. If the length of the solid is 3x and the width of the solid is x-2 Find the height?

Answer 1

`\mathsf{height}=x^2+x-9`

Explanation:

Volume of a cuboid = L x w x h

(where L is the length, w is the width and h is the height)

`\implies \mathsf{height=(volume)/(width * length) }`

Given:

`\mathsf{volume}=3x^4-3x^3-33x^2+54x`

`\mathsf{width}=x-2`

`\mathsf{length}=3x`

`\implies \mathsf{height}=(3x^4-3x^3-33x^2+54x)/(3x(x-2))`

Factor expression for volume

Factor out common term
`3x`:
`3x(x^3-x^2-11x+18)`

Factor
`x^3-x^2-11x+18`:
`3x(x-2)(x^2+x-9)`

`\implies \mathsf{height}=(3x(x-2)(x^2+x-9))/(3x(x-2))`

Cancel the common factors
`3x(x-2)`:

`\implies \mathsf{height}=x^2+x-9`