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.
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