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? (image of a cylinder with $h = 2x + 7$ and $r = x - 3$) options: $2\pi x^3 - 12\pi x^2 - 24\pi x + 63\pi$; $2\pi x^3 - 5\pi x^2 - 24\pi x + 63\pi$; (third option partially visible)

Explanation:

Step1: Substitute \( r = x - 3 \) and \( h = 2x + 7 \) into the volume formula \( V=\pi r^{2}h \)

We get \( V=\pi(x - 3)^{2}(2x + 7) \)

Step2: Expand \( (x - 3)^{2} \)

Using the formula \( (a - b)^{2}=a^{2}-2ab + b^{2} \), where \( a = x \) and \( b = 3 \), we have \( (x - 3)^{2}=x^{2}-6x + 9 \)
So now \( V=\pi(x^{2}-6x + 9)(2x + 7) \)

Step3: Multiply \( (x^{2}-6x + 9) \) and \( (2x + 7) \)

Using the distributive property (FOIL method extended):
\[

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

\]

Step4: Multiply by \( \pi \)

We get \( V=\pi(2x^{3}-5x^{2}-24x + 63)=2\pi x^{3}-5\pi x^{2}-24\pi x + 63\pi \)

Answer:

\( 2\pi x^{3}-5\pi x^{2}-24\pi x + 63\pi \) (corresponding to the option \( 2\pi x^{3}-5\pi x^{2}-24\pi x + 63\pi \))