To find the volume of a frustum formed by removing a smaller cone from a larger cone, we will follow these steps:
1. Determine the volume of the large cone:
The formula for the volume of a cone is:
[tex]\[
V = \frac{1}{3} \pi r^2 h
\][/tex]
where [tex]\( r \)[/tex] is the radius of the base and [tex]\( h \)[/tex] is the height.
Suppose the radius and height of the large cone are [tex]\( r_{large} \)[/tex] and [tex]\( h_{large} \)[/tex] respectively. Then the volume [tex]\( V_{large} \)[/tex] is:
[tex]\[
V_{large} = \frac{1}{3} \pi r_{large}^2 h_{large}
\][/tex]
2. Determine the volume of the small cone:
Suppose the radius and height of the small cone are [tex]\( r_{small} \)[/tex] and [tex]\( h_{small} \)[/tex] respectively. Then the volume [tex]\( V_{small} \)[/tex] is:
[tex]\[
V_{small} = \frac{1}{3} \pi r_{small}^2 h_{small}
\][/tex]
3. Volume of the frustum:
The volume of the frustum is obtained by subtracting the volume of the small cone from the volume of the large cone:
[tex]\[
V_{frustum} = V_{large} - V_{small}
\][/tex]
Substituting the formulas for the volumes of the large and small cones:
[tex]\[
V_{frustum} = \left( \frac{1}{3} \pi r_{large}^2 h_{large} \right) - \left( \frac{1}{3} \pi r_{small}^2 h_{small} \right)
\][/tex]
Factor out [tex]\(\frac{1}{3} \pi\)[/tex]:
[tex]\[
V_{frustum} = \frac{1}{3} \pi \left( r_{large}^2 h_{large} - r_{small}^2 h_{small} \right)
\][/tex]
Therefore, the volume of the frustum in terms of [tex]\(\pi\)[/tex] is:
[tex]\[
V_{frustum} = \frac{1}{3} \pi \left( r_{large}^2 h_{large} - r_{small}^2 h_{small} \right) \, \text{cm}^3
\][/tex]
