Question

the volume of a cylinder is given by the formula $v = \pi r^{2}h$, where $r$ is the radius of the cylinder and $h$ is the height.
which expression represents the volume of this cylinder?
$h = 2x + 2$
$r = x - 3$

Explanation:

Step1: Identify given values

Radius $r = x-3$, height $h = 2x+7$, volume formula $V=\pi r^2 h$

Step2: Substitute r into $r^2$

$$\begin{align*} r^2&=(x-3)^2\\ &=x^2 - 6x + 9 \end{align*}$$

Step3: Multiply by height h

$$\begin{align*} r^2h&=(x^2 - 6x + 9)(2x+7)\\ &=x^2(2x+7)-6x(2x+7)+9(2x+7)\\ &=2x^3+7x^2-12x^2-42x+18x+63\\ &=2x^3-5x^2-24x+63 \end{align*}$$

Step4: Multiply by $\pi$

$V=\pi(2x^3-5x^2-24x+63)$

Answer:

$\pi(2x^3 - 5x^2 - 24x + 63)$ or expanded as $2\pi x^3 - 5\pi x^2 - 24\pi x + 63\pi$