Question
A solid large cylinder has a radius of 4 cm and a height of 10 cm. A smaller cylinder with a radius of 2 cm and the same height is removed from the center of the larger cylinder, leaving a hollow space. What is the volume of the hollow space?
Answers
GPT-4o mini
To find the volume of the hollow space created by removing the smaller cylinder from the larger cylinder, we can follow these steps:
1. **Calculate the volume of the larger cylinder.** The volume \( V \) of a cylinder is given by the formula:
\[
V = \pi r^2 h
\]
For the larger cylinder with a radius of \( 4 \) cm and a height of \( 10 \) cm:
\[
V_{\text{large}} = \pi (4^2)(10) = \pi (16)(10) = 160\pi \, \text{cm}^3
\]
2. **Calculate the volume of the smaller cylinder.** Using the same formula for volume:
For the smaller cylinder with a radius of \( 2 \) cm and a height of \( 10 \) cm:
\[
V_{\text{small}} = \pi (2^2)(10) = \pi (4)(10) = 40\pi \, \text{cm}^3
\]
3. **Calculate the volume of the hollow space.** The volume of the hollow space is the volume of the larger cylinder minus the volume of the smaller cylinder:
\[
V_{\text{hollow}} = V_{\text{large}} - V_{\text{small}} = 160\pi - 40\pi = 120\pi \, \text{cm}^3
\]
Therefore, the volume of the hollow space is
\[
\boxed{120\pi} \, \text{cm}^3
\]
If needed, this can be approximated as
\[
120\pi \approx 376.99 \, \text{cm}^3 \quad (\text{using } \pi \approx 3.14)
\]
But it's most accurate to leave it as \( 120\pi \, \text{cm}^3 \).
1. **Calculate the volume of the larger cylinder.** The volume \( V \) of a cylinder is given by the formula:
\[
V = \pi r^2 h
\]
For the larger cylinder with a radius of \( 4 \) cm and a height of \( 10 \) cm:
\[
V_{\text{large}} = \pi (4^2)(10) = \pi (16)(10) = 160\pi \, \text{cm}^3
\]
2. **Calculate the volume of the smaller cylinder.** Using the same formula for volume:
For the smaller cylinder with a radius of \( 2 \) cm and a height of \( 10 \) cm:
\[
V_{\text{small}} = \pi (2^2)(10) = \pi (4)(10) = 40\pi \, \text{cm}^3
\]
3. **Calculate the volume of the hollow space.** The volume of the hollow space is the volume of the larger cylinder minus the volume of the smaller cylinder:
\[
V_{\text{hollow}} = V_{\text{large}} - V_{\text{small}} = 160\pi - 40\pi = 120\pi \, \text{cm}^3
\]
Therefore, the volume of the hollow space is
\[
\boxed{120\pi} \, \text{cm}^3
\]
If needed, this can be approximated as
\[
120\pi \approx 376.99 \, \text{cm}^3 \quad (\text{using } \pi \approx 3.14)
\]
But it's most accurate to leave it as \( 120\pi \, \text{cm}^3 \).