Question

question 5 (multiple - choice, worth 2 points) (05.01 mc) a cone has a volume of 144π in³ and a diameter of 12 in. wilson states that a cylinder with the same height and diameter has the same volume. which statement explains whether or not wilson is correct? a a cylinder in which h = 12 and d = 12 has a volume of 144π in³, therefore, wilson is correct. b a cylinder in which h = 4 and d = 12 has a volume of 432π in³, therefore, wilson is incorrect. c a cylinder in which h = 12 and d = 12 has a volume of 432π in³, therefore, wilson is incorrect. d a cylinder in which h = 4 and d = 12 has a volume of 144π in³, therefore, wilson is correct.

Explanation:

Step1: Recall volume formulas

The volume formula for a cone is $V_{cone}=\frac{1}{3}\pi r^{2}h$ and for a cylinder is $V_{cylinder}=\pi r^{2}h$. Given the cone has $V = 144\pi$ in³ and $d = 12$ in (so $r=6$ in). Using the cone - volume formula $144\pi=\frac{1}{3}\pi(6)^{2}h$. Solving for $h$:
\[

$$\begin{align*} 144\pi&=\frac{1}{3}\pi\times36\times h\\ 144& = 12h\\ h&=12\text{ in} \end{align*}$$

\]

Step2: Calculate cylinder volume

For a cylinder with $r = 6$ in and $h = 12$ in, using the cylinder - volume formula $V_{cylinder}=\pi r^{2}h=\pi\times(6)^{2}\times12=\pi\times36\times12 = 432\pi$ in³.

Answer:

A cylinder in which $h = 12$ and $d = 12$ has a volume of $432\pi$ in³, therefore, Wilson is incorrect.