Question
Find the radius of a cone with a volume of 3419.46ft cubed and that has a height of 27 feet
Answers
Answered by
GPT-4o mini
The volume \( V \) of a cone can be calculated using the formula:
\[
V = \frac{1}{3} \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height of the cone. We can rearrange this formula to solve for the radius \( r \):
1. Start with the volume formula:
\[
3419.46 = \frac{1}{3} \pi r^2 (27)
\]
2. To simplify, first multiply both sides by 3:
\[
3 \times 3419.46 = \pi r^2 (27)
\]
\[
10258.38 = 27 \pi r^2
\]
3. Next, divide both sides by \( 27\pi \):
\[
\frac{10258.38}{27\pi} = r^2
\]
4. Now calculate \( r^2 \):
\[
r^2 = \frac{10258.38}{27 \times 3.14159} \approx \frac{10258.38}{84.823} \approx 120.81
\]
5. Finally, take the square root of both sides to find \( r \):
\[
r \approx \sqrt{120.81} \approx 10.98
\]
Thus, the radius of the cone is approximately \( 10.98 \) feet.
\[
V = \frac{1}{3} \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height of the cone. We can rearrange this formula to solve for the radius \( r \):
1. Start with the volume formula:
\[
3419.46 = \frac{1}{3} \pi r^2 (27)
\]
2. To simplify, first multiply both sides by 3:
\[
3 \times 3419.46 = \pi r^2 (27)
\]
\[
10258.38 = 27 \pi r^2
\]
3. Next, divide both sides by \( 27\pi \):
\[
\frac{10258.38}{27\pi} = r^2
\]
4. Now calculate \( r^2 \):
\[
r^2 = \frac{10258.38}{27 \times 3.14159} \approx \frac{10258.38}{84.823} \approx 120.81
\]
5. Finally, take the square root of both sides to find \( r \):
\[
r \approx \sqrt{120.81} \approx 10.98
\]
Thus, the radius of the cone is approximately \( 10.98 \) feet.
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