Question
What is the ratio of the volume of a cone to the volume of a cylinder with the same base and height?
Answers
GPT-4o mini
To find the ratio of the volume of a cone to the volume of a cylinder that have the same base radius \( r \) and height \( h \), we can follow the formulas for the volumes of both solids.
1. The volume \( V_c \) of a cone is given by the formula:
\[
V_c = \frac{1}{3} \pi r^2 h
\]
2. The volume \( V_{\text{cyl}} \) of a cylinder is given by the formula:
\[
V_{\text{cyl}} = \pi r^2 h
\]
Now, we want to find the ratio of the volume of the cone to the volume of the cylinder:
\[
\text{Ratio} = \frac{V_c}{V_{\text{cyl}}} = \frac{\frac{1}{3} \pi r^2 h}{\pi r^2 h}
\]
In this expression, \( \pi r^2 h \) (the volume of the cylinder) in the denominator cancels out, leading to:
\[
\text{Ratio} = \frac{1}{3}
\]
Thus, the ratio of the volume of a cone to the volume of a cylinder with the same base and height is:
\[
\frac{1}{3}
\]
1. The volume \( V_c \) of a cone is given by the formula:
\[
V_c = \frac{1}{3} \pi r^2 h
\]
2. The volume \( V_{\text{cyl}} \) of a cylinder is given by the formula:
\[
V_{\text{cyl}} = \pi r^2 h
\]
Now, we want to find the ratio of the volume of the cone to the volume of the cylinder:
\[
\text{Ratio} = \frac{V_c}{V_{\text{cyl}}} = \frac{\frac{1}{3} \pi r^2 h}{\pi r^2 h}
\]
In this expression, \( \pi r^2 h \) (the volume of the cylinder) in the denominator cancels out, leading to:
\[
\text{Ratio} = \frac{1}{3}
\]
Thus, the ratio of the volume of a cone to the volume of a cylinder with the same base and height is:
\[
\frac{1}{3}
\]