Question

The volume of the base of a rectangular prism is given by 12x ^ 3 + 23x ^ 2 + 46x + 45 The height of the rectangular prism is given by 4x + 5 Write an expression for the area of the base of the rectangular prism.

Answer 1

`Base\ Area=3x^2 + 2x + 9`

Explanation:

Given

`Volume = 12x^3 + 23x^2 + 46x + 45`

`Height = 4x + 5`

Required

Find the base area

Volume is calculated as:

`Volume = Height * Base\ Area`

Make Area the subject

`Base\ Area=(Volume)/(Height)`

Substitute values for Height and Volume

`Base\ Area=(12x^3 + 23x^2 + 46x + 45)/(4x + 5)`

Expand the numerator

`Base\ Area=(12x^3 + 8x^2 + 15x^2 + 36x + 10x + 45)/(4x + 5)`

Rearrange

`Base\ Area=(12x^3 + 8x^2 + 36x + 15x^2 + 10x + 45)/(4x + 5)`

Factorize

`Base\ Area=(4x(3x^2 + 2x + 9) +5(3x^2 + 2x + 9))/(4x + 5)`

`Base\ Area=((4x +5)(3x^2 + 2x + 9))/(4x + 5)`

Cancel out 4x + 5

`Base\ Area=(3x^2 + 2x + 9)`

`Base\ Area=3x^2 + 2x + 9`