Question

Find the volume of the cone with slant height 61 feet and radius of 11 feet
The height of the cone is______ feet.
The volume of the cone is______ cubic feet.

Answer 1

To find the height and volume of the cone, we can use the following formulas and relationships.

1. **Finding the height** of the cone:
We know that the slant height (l), radius (r), and height (h) of a cone are related by the Pythagorean theorem:
\[
l^2 = r^2 + h^2
\]
Given:
- Slant height \( l = 61 \) feet
- Radius \( r = 11 \) feet

Plugging in the values:
\[
61^2 = 11^2 + h^2
\]
\[
3721 = 121 + h^2
\]
\[
h^2 = 3721 - 121 = 3600
\]
\[
h = \sqrt{3600} = 60 \text{ feet}
\]

2. **Finding the volume** of the cone:
The volume V of a cone can be calculated using the formula:
\[
V = \frac{1}{3} \pi r^2 h
\]
Substituting the known values:
\[
V = \frac{1}{3} \pi (11^2)(60)
\]
\[
= \frac{1}{3} \pi (121)(60)
\]
\[
= \frac{1}{3} \pi (7260)
\]
\[
= 2420 \pi \text{ cubic feet}
\]

Using the approximate value of \(\pi \approx 3.14\):
\[
V \approx 2420 \times 3.14 \approx 7605.8 \text{ cubic feet}
\]

Thus, we have:
- The height of the cone is **60 feet**.
- The volume of the cone is approximately **7605.8 cubic feet** (or **2420π cubic feet**).