Question
Drag the points to create two different cylinders with the same volume. You can change both the height and radius of each cylinder.
What is the volume of the cylinders? give me an exsample
What is the volume of the cylinders? give me an exsample
Answers
GPT-4o mini
The volume \( V \) of a cylinder can be calculated using the formula:
\[
V = \pi r^2 h
\]
where:
- \( V \) is the volume,
- \( r \) is the radius of the base of the cylinder,
- \( h \) is the height of the cylinder.
To create two different cylinders with the same volume, you can vary the height and radius while keeping the product \( r^2 h \) constant.
### Example:
1. **Cylinder 1:**
- Radius \( r_1 = 3 \) units
- Height \( h_1 = 4 \) units
Volume:
\[
V_1 = \pi (3)^2 (4) = \pi (9)(4) = 36\pi \text{ cubic units}
\]
2. **Cylinder 2:**
- Radius \( r_2 = 6 \) units
- Height \( h_2 = 2 \) units
Volume:
\[
V_2 = \pi (6)^2 (2) = \pi (36)(2) = 72\pi \text{ cubic units}
\]
### Finding Equivalent Volumes:
You can create a scenario where the volumes are equal. Let's say you want both cylinders to have a volume of \( 36\pi \) cubic units:
- **Cylinder 1:**
- Radius \( r_1 = 3 \) units
- Height \( h_1 = 4 \) units
- Volume = \( 36\pi \)
- **Cylinder 2:**
- Radius \( r_2 = 4 \) units
- Height \( h_2 = \frac{36}{\pi (4^2)} = \frac{36}{16\pi} = 2.25 \) units
- Volume = \( 36\pi \)
So, both cylinders have the same volume of \( 36\pi \) cubic units, but different dimensions. You can continue to adjust other combinations of radius and height to find more pairs of cylinders with equal volume.
\[
V = \pi r^2 h
\]
where:
- \( V \) is the volume,
- \( r \) is the radius of the base of the cylinder,
- \( h \) is the height of the cylinder.
To create two different cylinders with the same volume, you can vary the height and radius while keeping the product \( r^2 h \) constant.
### Example:
1. **Cylinder 1:**
- Radius \( r_1 = 3 \) units
- Height \( h_1 = 4 \) units
Volume:
\[
V_1 = \pi (3)^2 (4) = \pi (9)(4) = 36\pi \text{ cubic units}
\]
2. **Cylinder 2:**
- Radius \( r_2 = 6 \) units
- Height \( h_2 = 2 \) units
Volume:
\[
V_2 = \pi (6)^2 (2) = \pi (36)(2) = 72\pi \text{ cubic units}
\]
### Finding Equivalent Volumes:
You can create a scenario where the volumes are equal. Let's say you want both cylinders to have a volume of \( 36\pi \) cubic units:
- **Cylinder 1:**
- Radius \( r_1 = 3 \) units
- Height \( h_1 = 4 \) units
- Volume = \( 36\pi \)
- **Cylinder 2:**
- Radius \( r_2 = 4 \) units
- Height \( h_2 = \frac{36}{\pi (4^2)} = \frac{36}{16\pi} = 2.25 \) units
- Volume = \( 36\pi \)
So, both cylinders have the same volume of \( 36\pi \) cubic units, but different dimensions. You can continue to adjust other combinations of radius and height to find more pairs of cylinders with equal volume.