To find the volume of a cone, we can use the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( V \) is the volume, \( r \) is the radius, and \( h \) is the height.
Step 1: Find the height of the cone.
We are given the slant height (\( l \)) and the radius (\( r \)). We can use the Pythagorean theorem to find the height (\( h \)) of the cone. The relationship between the radius, height, and slant height can be defined as:
\[ l^2 = r^2 + h^2 \]
Given:
- Slant height \( l = 29 \) feet
- Radius \( r = 20 \) feet
We can rearrange the equation to solve for \( h \):
\[ h^2 = l^2 - r^2 \]
Now, plug in the values:
\[ h^2 = 29^2 - 20^2 \]
Calculating \( 29^2 \) and \( 20^2 \):
\[ 29^2 = 841 \] \[ 20^2 = 400 \]
So,
\[ h^2 = 841 - 400 = 441 \]
Now, take the square root to find \( h \):
\[ h = \sqrt{441} = 21 \text{ feet} \]
Step 2: Calculate the volume of the cone.
Now that we have the height, we can substitute the radius and height into the volume formula:
\[ V = \frac{1}{3} \pi r^2 h \] \[ V = \frac{1}{3} \times 3.14 \times 20^2 \times 21 \]
Calculating \( 20^2 \):
\[ 20^2 = 400 \]
Now substitute this back into the volume formula:
\[ V = \frac{1}{3} \times 3.14 \times 400 \times 21 \] \[ V = \frac{1}{3} \times 3.14 \times 8400 \]
Calculate \( 3.14 \times 8400 \):
\[ 3.14 \times 8400 = 26376 \]
Now divide by 3:
\[ V = \frac{26376}{3} = 8792 \]
Final Answer
The volume of the cone is approximately 8792 cubic feet.