Question
Use the relationship among the formulas for the volumes of cones, cylinders, and spheres to solve the following problem. The volume of a cylinder is 36 cm3
. What is the volume of a sphere if its radius is the same as the cylinder’s and the height of the cylinder is equal to the sphere’s diameter?(1 point)
Responses
54 cm3
54 cm cubed
12 cm3
12 cm cubed
18 cm3
18 cm cubed
24 cm3
. What is the volume of a sphere if its radius is the same as the cylinder’s and the height of the cylinder is equal to the sphere’s diameter?(1 point)
Responses
54 cm3
54 cm cubed
12 cm3
12 cm cubed
18 cm3
18 cm cubed
24 cm3
Answers
GPT-4o mini
To solve the problem, we need to relate the volumes of the cylinder and the sphere based on their dimensions.
1. **Volume of a cylinder**: The formula for the volume of a cylinder is given by:
\[
V_{\text{cylinder}} = \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height.
2. **Volume of a sphere**: The formula for the volume of a sphere is:
\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\]
Given that the volume of the cylinder is \( 36 \, \text{cm}^3 \), we can set up the equation for the cylinder:
\[
\pi r^2 h = 36
\]
We also know the height of the cylinder is equal to the diameter of the sphere. Since the radius \( r \) of the sphere is the same as that of the cylinder, the diameter of the sphere is \( 2r \).
Thus, we can state that:
\[
h = 2r
\]
Substituting \( h \) in the cylinder volume equation:
\[
\pi r^2 (2r) = 36
\]
This simplifies to:
\[
2\pi r^3 = 36
\]
Dividing both sides by \( 2\pi \):
\[
r^3 = \frac{36}{2\pi} = \frac{18}{\pi}
\]
We now substitute this value of \( r \) into the formula for the volume of the sphere:
\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\]
Substituting \( r^3 = \frac{18}{\pi} \):
\[
V_{\text{sphere}} = \frac{4}{3} \pi \left(\frac{18}{\pi}\right)
\]
The \( \pi \) terms cancel out, leaving:
\[
V_{\text{sphere}} = \frac{4}{3} \times 18 = \frac{72}{3} = 24
\]
Thus, the volume of the sphere is:
\[
\boxed{24 \, \text{cm}^3}
\]
1. **Volume of a cylinder**: The formula for the volume of a cylinder is given by:
\[
V_{\text{cylinder}} = \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height.
2. **Volume of a sphere**: The formula for the volume of a sphere is:
\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\]
Given that the volume of the cylinder is \( 36 \, \text{cm}^3 \), we can set up the equation for the cylinder:
\[
\pi r^2 h = 36
\]
We also know the height of the cylinder is equal to the diameter of the sphere. Since the radius \( r \) of the sphere is the same as that of the cylinder, the diameter of the sphere is \( 2r \).
Thus, we can state that:
\[
h = 2r
\]
Substituting \( h \) in the cylinder volume equation:
\[
\pi r^2 (2r) = 36
\]
This simplifies to:
\[
2\pi r^3 = 36
\]
Dividing both sides by \( 2\pi \):
\[
r^3 = \frac{36}{2\pi} = \frac{18}{\pi}
\]
We now substitute this value of \( r \) into the formula for the volume of the sphere:
\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\]
Substituting \( r^3 = \frac{18}{\pi} \):
\[
V_{\text{sphere}} = \frac{4}{3} \pi \left(\frac{18}{\pi}\right)
\]
The \( \pi \) terms cancel out, leaving:
\[
V_{\text{sphere}} = \frac{4}{3} \times 18 = \frac{72}{3} = 24
\]
Thus, the volume of the sphere is:
\[
\boxed{24 \, \text{cm}^3}
\]