Question

The radius of the cone is 7 in and y = 25 in. What is the volume of the cone in terms of π?

A cone with a right triangle formed from its dimensions; the value of the height is h, and the value of the slant height is y; the height x and the radius form a right angle at the center of the cone.

392π in3
408π in3
112π in3
117π in3

Answer 1

To find the volume of a cone, you can use the formula:

\[
V = \frac{1}{3} \pi r^2 h
\]

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

From the information provided:
- The radius \( r = 7 \) in
- The slant height \( y = 25 \) in

We need to first find the height \( h \) of the cone. We can use the Pythagorean theorem, as the radius, height, and slant height form a right triangle:

\[
y^2 = r^2 + h^2
\]

Substituting the values of \( y \) and \( r \):

\[
25^2 = 7^2 + h^2
\]

Calculating \( 25^2 \) and \( 7^2 \):

\[
625 = 49 + h^2
\]

Subtract 49 from both sides:

\[
625 - 49 = h^2
\]

\[
576 = h^2
\]

Taking the square root of both sides gives:

\[
h = \sqrt{576} = 24 \text{ in}
\]

Now we have:

- \( r = 7 \) in
- \( h = 24 \) in

Now we can calculate the volume:

\[
V = \frac{1}{3} \pi (7^2)(24)
\]
\[
V = \frac{1}{3} \pi (49)(24)
\]
\[
V = \frac{1}{3} \pi (1176)
\]
\[
V = 392 \pi \text{ in}^3
\]

Thus, the volume of the cone is \(\boxed{392\pi}\) in³.