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:
-
Cylinder 1:
- Radius \( r_1 = 3 \) units
- Height \( h_1 = 4 \) units
Volume: \[ V_1 = \pi (3)^2 (4) = \pi (9)(4) = 36\pi \text{ cubic units} \]
-
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.