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

Answer:

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

Step-by-step 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

$Base\ Area=\frac{Volume}{Height}$

Substitute values for Height and Volume

$Base\ Area=\frac{12x^3 + 23x^2 + 46x + 45}{4x + 5}$

Expand the numerator

$Base\ Area=\frac{12x^3 + 8x^2 + 15x^2 + 36x + 10x + 45}{4x + 5}$

Rearrange

$Base\ Area=\frac{12x^3 + 8x^2 + 36x + 15x^2 + 10x + 45}{4x + 5}$

Factorize

$Base\ Area=\frac{4x(3x^2 + 2x + 9) +5(3x^2 + 2x + 9)}{4x + 5}$

$Base\ Area=\frac{(4x +5)(3x^2 + 2x + 9)}{4x + 5}$

Cancel out 4x + 5

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

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