Chapter 8.8, Problem 44PPE

(a)

To determine

To factorize:the expression $2\pi x^3+12\pi x^2+18\pi x$

Answer to Problem 44PPE

The factors of $2\pi x^3+12\pi x^2+18\pi x$ is $2\pi x{\left(x+3\right)}^2$

Explanation of Solution

Given Information:The polynomial $2\pi x^3+12\pi x^2+18\pi x$ represents the volume of a cylinder.

The volume $\text{V}$ of cylinder with radius $r$ and height $h$ is given by $\pi r^{2}h$ .

Concept used:

To factorise $2\pi x^3+12\pi x^2+18\pi x$ take out the common terms and use the formula.

Formula used:

  ${\left(a+b\right)}^2=a^2+b^2+2ab$

Calculations:

  $\begin{array}{c}2\pi x^3+12\pi x^2+18\pi x=2\pi x\left(x^2+6x+9\right)\\ =2\pi x\left(x^2+2\left(3\right)x+3^2\right)\\ =2\pi x{\left(x+3\right)}^2\end{array}$

(b)

To determine

the expression for the possible radius of the cylinder.

Answer to Problem 44PPE

The expression with radius $r$ is $\left(x+3\right)$

Explanation of Solution

Given Information:The polynomial $2\pi x^3+12\pi x^2+18\pi x$ represents the volume of a cylinder.

The volume $\text{V}$ of cylinder with radius $r$ and height $h$ is given by $\pi r^{2}h$

Concept used:

Compare $2\pi x{\left(x+3\right)}^2$ with the formula of cylinder.

Formula used:

The volume $\text{V}$ of cylinder with radius $r$ and height $h$ is given by $\pi r^{2}h$

Calculations:

Compare $2\pi x{\left(x+3\right)}^2$ with $\pi r^{2}h$ .

Rewrite in the form $\pi {\left(x+3\right)}^2\left(2x\right)$

Therefore, the expression with radius $r$ is $\left(x+3\right)$