Question
The volume of a cone is 500 cm3
and the height is approximately 13.5 cm, what is the approximate radius of the cone?
and the height is approximately 13.5 cm, what is the approximate radius of the cone?
Answers
GPT-4o mini
To find the radius of a cone given its volume and height, you can use the formula for the volume of a cone:
\[
V = \frac{1}{3} \pi r^2 h
\]
where \( V \) is the volume, \( r \) is the radius, and \( h \) is the height.
In this case, you know that the volume \( V = 500 \) cm³ and the height \( h \approx 13.5 \) cm. You want to find the radius \( r \).
1. Substitute the known values into the formula:
\[
500 = \frac{1}{3} \pi r^2 (13.5)
\]
2. Multiply both sides by 3 to eliminate the fraction:
\[
1500 = \pi r^2 (13.5)
\]
3. Divide both sides by \( 13.5\pi \):
\[
r^2 = \frac{1500}{13.5 \pi}
\]
4. Calculate \( r^2 \):
First, calculate \( 13.5 \pi \):
\[
13.5 \pi \approx 13.5 \times 3.14159 \approx 42.411
\]
Now divide 1500 by \( 42.411 \):
\[
r^2 \approx \frac{1500}{42.411} \approx 35.34
\]
5. Take the square root to find \( r \):
\[
r \approx \sqrt{35.34} \approx 5.95
\]
Thus, the approximate radius of the cone is \( r \approx 5.95 \) cm.
\[
V = \frac{1}{3} \pi r^2 h
\]
where \( V \) is the volume, \( r \) is the radius, and \( h \) is the height.
In this case, you know that the volume \( V = 500 \) cm³ and the height \( h \approx 13.5 \) cm. You want to find the radius \( r \).
1. Substitute the known values into the formula:
\[
500 = \frac{1}{3} \pi r^2 (13.5)
\]
2. Multiply both sides by 3 to eliminate the fraction:
\[
1500 = \pi r^2 (13.5)
\]
3. Divide both sides by \( 13.5\pi \):
\[
r^2 = \frac{1500}{13.5 \pi}
\]
4. Calculate \( r^2 \):
First, calculate \( 13.5 \pi \):
\[
13.5 \pi \approx 13.5 \times 3.14159 \approx 42.411
\]
Now divide 1500 by \( 42.411 \):
\[
r^2 \approx \frac{1500}{42.411} \approx 35.34
\]
5. Take the square root to find \( r \):
\[
r \approx \sqrt{35.34} \approx 5.95
\]
Thus, the approximate radius of the cone is \( r \approx 5.95 \) cm.