Question

a cone has a volume of 9π in³ and a diameter of 6 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 cylinder in which h = 1 and d = 6 has a volume of 27π in³, therefore, wilson is incorrect
a cylinder in which h = 3 and d = 6 has a volume of 27π in³, therefore, wilson is incorrect
a cylinder in which h = 1 and d = 6 has a volume of 9π in³, therefore, wilson is correct
a cylinder in which h = 3 and d = 6 has a volume of 9π in³, therefore, wilson is correct

Explanation:

Step1: Recall volume formulas

The volume of a cone is \( V_{cone}=\frac{1}{3}\pi r^{2}h \), and the volume of a cylinder is \( V_{cylinder}=\pi r^{2}h \), where \( r \) is the radius and \( h \) is the height. Given the diameter \( d = 6 \) in, the radius \( r=\frac{d}{2}=3 \) in. The cone has a volume of \( 9\pi\) \( \text{in}^3 \). Using the cone volume formula: \( 9\pi=\frac{1}{3}\pi(3)^{2}h \). Solving for \( h \):
\[

$$\begin{align*} 9\pi&=\frac{1}{3}\pi\times9\times h\\ 9\pi& = 3\pi h\\ h&= 3 \end{align*}$$

\]
So the height of the cone (and the cylinder, since they have the same height) is \( h = 3 \) in.

Step2: Calculate cylinder volume

Now, calculate the volume of the cylinder with \( r = 3 \) in and \( h=3 \) in using the cylinder volume formula \( V_{cylinder}=\pi r^{2}h \):
\[
V_{cylinder}=\pi\times(3)^{2}\times3=\pi\times9\times3 = 27\pi\ \text{in}^3
\]
The cone's volume is \( 9\pi \), and the cylinder's volume is \( 27\pi \), which is three times the cone's volume. So Wilson is incorrect. Now check the options:

• Option 1: \( h = 1 \), \( d = 6 \) (so \( r = 3 \)). Volume of cylinder: \( \pi\times3^{2}\times1=9\pi \), but we found \( h = 3 \), so this is wrong.
• Option 2: \( h = 3 \), \( d = 6 \) ( \( r = 3 \) ). Volume: \( \pi\times3^{2}\times3 = 27\pi \), so Wilson is incorrect. This matches.
• Option 3: Cylinder volume here is \( 9\pi \), but we know it should be \( 27\pi \) when \( h = 3 \), so Wilson is not correct.
• Option 4: Cylinder volume here is said to be \( 9\pi \), but actual is \( 27\pi \), so wrong.

Answer:

A cylinder in which \( h = 3 \) and \( d = 6 \) has a volume of \( 27\pi\ \text{in}^3 \), therefore, Wilson is incorrect (the second option in the list, i.e., the option with "A cylinder in which \( h = 3 \) and \( d = 6 \) has a volume of \( 27\pi\ \text{in}^3 \), therefore, Wilson is incorrect")