Question

Jeremy must find the volume of the shaded body formed when a cylinder of radius 2 is removed from a cylinder of radius 5 as pictured below.

A cylinder of radius 2 is removed from a cylinder of radius 5. The height of the cylinders is 18.

If the height of the cylinders is 18 units, which expression is equivalent to the volume of the shaded part of the figure in cubic units?
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mc015-5.jpg

Answer 1

To find the volume of the shaded body formed when a cylinder of radius 2 is removed from a cylinder of radius 5, we can use the formula for the volume of a cylinder, which is given by:

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

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

1. **Calculate the volume of the larger cylinder (radius 5)**:
- Radius = 5
- Height = 18
\[
V_{large} = \pi (5^2) (18) = \pi (25) (18) = 450\pi
\]

2. **Calculate the volume of the smaller cylinder (radius 2)**:
- Radius = 2
- Height = 18
\[
V_{small} = \pi (2^2) (18) = \pi (4) (18) = 72\pi
\]

3. **Find the volume of the shaded part**:
- Volume of the shaded part is the volume of the larger cylinder minus the volume of the smaller cylinder:
\[
V_{shaded} = V_{large} - V_{small} = 450\pi - 72\pi = 378\pi
\]

Thus, the expression equivalent to the volume of the shaded part of the figure is:

\[
378\pi
\]

in cubic units.