To find the volume of the cone given the radius and the slant height, let's go through the steps methodically:
1. Identify the given values:
- Radius (r) = 7 inches
- Slant height (y) = 25 inches
2. Determine the height (h) of the cone:
The height of the cone can be found using the Pythagorean theorem in the right triangle formed by the radius, the height, and the slant height. We know:
[tex]\[
y^2 = r^2 + h^2
\][/tex]
Substituting the given values:
[tex]\[
25^2 = 7^2 + h^2
\][/tex]
[tex]\[
625 = 49 + h^2
\][/tex]
Solving for [tex]\( h \)[/tex]:
[tex]\[
h^2 = 625 - 49
\][/tex]
[tex]\[
h^2 = 576
\][/tex]
[tex]\[
h = \sqrt{576}
\][/tex]
[tex]\[
h = 24
\][/tex]
3. Calculate the volume (V) of the cone:
The formula for the volume of a cone is given by:
[tex]\[
V = \frac{1}{3} \pi r^2 h
\][/tex]
Substituting the values for the radius and height:
[tex]\[
V = \frac{1}{3} \pi (7^2) (24)
\][/tex]
[tex]\[
V = \frac{1}{3} \pi (49) (24)
\][/tex]
[tex]\[
V = \frac{1}{3} \pi (1176)
\][/tex]
[tex]\[
V = 392 \pi
\][/tex]
Therefore, the volume of the cone in terms of π is [tex]\( 392 \pi \)[/tex] cubic inches.
