Question

Left cylinder
height 8
radius 8
Right cylinder
height 4
radius 8

Drag the points to create two different cylinders with the same volume. You can change both the height and radius of earth cylinder.

What is the volume of the cylinders?

Answer 1

The volume \( V \) of a cylinder can be calculated using the formula:

\[
V = \pi r^2 h
\]

where \( r \) is the radius and \( h \) is the height.

### Left Cylinder
For the left cylinder:
- Height \( h_1 = 8 \)
- Radius \( r_1 = 8 \)

Calculating the volume:

\[
V_1 = \pi (8^2)(8) = \pi (64)(8) = \pi (512) = 512\pi
\]

### Right Cylinder
For the right cylinder:
- Height \( h_2 = 4 \)
- Radius \( r_2 = 8 \)

Calculating the volume:

\[
V_2 = \pi (8^2)(4) = \pi (64)(4) = \pi (256) = 256\pi
\]

### Summary of Volumes
- Volume of Left Cylinder: \( 512\pi \)
- Volume of Right Cylinder: \( 256\pi \)

### Creating Two Different Cylinders With the Same Volume
If you want to create two different cylinders with the same volume, you'll need to select different values for height and radius such that:

\[
\pi r_1^2 h_1 = \pi r_2^2 h_2
\]

This simplifies to:

\[
r_1^2 h_1 = r_2^2 h_2
\]

You can choose different combinations of radius and height to achieve equal volumes. For example:

If you want both cylinders to have a volume of \( 256\pi \) (the volume of the right cylinder), you can select different values for radius and height. For instance:

1. **Cylinder A:**
- Height: \( 8 \)
- Volume: \( 256\pi \)
- Therefore, \( r \) needs to satisfy \( r^2(8) = 256 \) which gives \( r^2 = 32 \) or \( r = 4\sqrt{2} \approx 5.66 \).

2. **Cylinder B:**
- Height: \( 4 \)
- Volume: \( 256\pi \)
- Therefore, \( r^2(4) = 256 \), giving \( r^2 = 64 \) or \( r = 8 \).

These choices give you two different cylinders with the same volume: one taller and thinner and one shorter and wider. Feel free to adjust the dimensions to suit your needs!