Question
A cone and a cylinder have the same base radius an the same height. If the volume of the cone is 10 π cubic units, what is the volume of the cylinder?
Answers
GPT-4o mini
To find the volume of the cylinder given that the cone has a volume of \(10\pi\) cubic units, we start by recalling the formulas for the volumes of a cone and a cylinder.
The volume \(V\) of a cone is given by the formula:
\[
V = \frac{1}{3} \pi r^2 h
\]
where \(r\) is the radius of the base and \(h\) is the height.
The volume \(V\) of a cylinder is given by the formula:
\[
V = \pi r^2 h
\]
Since we know that the cone has a volume of \(10\pi\), we can set up the equation for the cone's volume:
\[
\frac{1}{3} \pi r^2 h = 10\pi
\]
We can simplify this equation by dividing both sides by \(\pi\):
\[
\frac{1}{3} r^2 h = 10
\]
Next, we multiply both sides by 3 to solve for \(r^2 h\):
\[
r^2 h = 30
\]
Now, we can use this value to find the volume of the cylinder:
\[
V = \pi r^2 h
\]
Substituting \(r^2 h = 30\) into the cylinder's volume formula:
\[
V = \pi (30) = 30\pi
\]
Thus, the volume of the cylinder is \(\boxed{30\pi}\) cubic units.
The volume \(V\) of a cone is given by the formula:
\[
V = \frac{1}{3} \pi r^2 h
\]
where \(r\) is the radius of the base and \(h\) is the height.
The volume \(V\) of a cylinder is given by the formula:
\[
V = \pi r^2 h
\]
Since we know that the cone has a volume of \(10\pi\), we can set up the equation for the cone's volume:
\[
\frac{1}{3} \pi r^2 h = 10\pi
\]
We can simplify this equation by dividing both sides by \(\pi\):
\[
\frac{1}{3} r^2 h = 10
\]
Next, we multiply both sides by 3 to solve for \(r^2 h\):
\[
r^2 h = 30
\]
Now, we can use this value to find the volume of the cylinder:
\[
V = \pi r^2 h
\]
Substituting \(r^2 h = 30\) into the cylinder's volume formula:
\[
V = \pi (30) = 30\pi
\]
Thus, the volume of the cylinder is \(\boxed{30\pi}\) cubic units.