Chapter 8.8, Problem 6STP

The polynomial $3\pi x^3+24\pi x^2+48\pi x$ represents the volume of a cylinder. The formula for the volume of a cylinder with radius $r$ and height $h$ is $V=\pi r^{2}h$.

a. Factor $3\pi x^3+24\pi x^2+48\pi x$.

b. Write a linear expression for a possible radius of the cylinder. Explain.

To determine

To factor the given expression.

Answer to Problem 6STP

  $3\pi x^3+24\pi x^2+48\pi x=3\pi x{\left(x+4\right)}^2$

Explanation of Solution

Given information :

  $3\pi x^3+24\pi x^2+48\pi x$

Now we shall factorize the given,

  $\begin{array}{c}3\pi x^3+24\pi x^2+48\pi x=3\pi x\left(x^2+8x+16\right)\\ =3\pi x\left(x^2+4x+4x+16\right)\\ =3\pi x\left(x\left(x+4\right)+4\left(x+4\right)\right)\\ =3\pi x\left(x+4\right)\left(x+4\right)\\ =3\pi x{\left(x+4\right)}^2\end{array}$

To determine

To determine a linear expression for a possible radius of cylinder.

Answer to Problem 6STP

  $r=x+4$

Explanation of Solution

Given information :

  $3\pi x^3+24\pi x^2+48\pi x$

In the first part we calculated $3\pi x^3+24\pi x^2+48\pi x=3\pi x{\left(x+4\right)}^2$

Formula given for volume is,

  $V=\pi r^{2}h$

Now we shall compare the factors and the coffiecients to determine the possible expression for radius.

Hence we get,

  $\pi r^{2}h=3\pi x{\left(x+4\right)}^2$

The only square term in both the sides is radius and $\left(x+4\right)$ .

Hence it can be determined that it can be the radius of the cylinder.

Hence $r=x+4$