Question

a cone has a volume of 1102.14 cubic millimeters and a height of 13 millimeters. what is its radius?
round your answer to the nearest hundredth.
r ≈ \square millimeters

Explanation:

Step1: Recall the volume formula for a cone

The volume \( V \) of a cone is given by the formula \( V = \frac{1}{3}\pi r^2 h \), where \( r \) is the radius and \( h \) is the height. We know \( V = 1102.14 \) cubic millimeters and \( h = 13 \) millimeters. We need to solve for \( r \).
First, we can rearrange the formula to solve for \( r^2 \):
Starting with \( V=\frac{1}{3}\pi r^2 h \), multiply both sides by 3: \( 3V=\pi r^2 h \)
Then divide both sides by \( \pi h \): \( r^2=\frac{3V}{\pi h} \)

Step2: Substitute the known values

Substitute \( V = 1102.14 \), \( h = 13 \) into the formula for \( r^2 \):
\( r^2=\frac{3\times1102.14}{\pi\times 13} \)
Calculate the numerator: \( 3\times1102.14 = 3306.42 \)
Calculate the denominator: \( \pi\times13\approx 40.8407 \)
Then \( r^2=\frac{3306.42}{40.8407}\approx 80.96 \)

Step3: Solve for \( r \)

Take the square root of both sides to find \( r \):
\( r=\sqrt{80.96}\approx 8.9978\approx 9.00 \) (rounded to the nearest hundredth)

Answer:

\( r\approx\boxed{9.00} \) millimeters