Question

To find the volume of the cone in the picture where S = 13 cm and R = 5 cm:



(3 points)
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Step 1
Step 2
Step 3

Answer 1

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

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

where:
- \( V \) is the volume,
- \( R \) is the radius of the base of the cone,
- \( h \) is the height of the cone.

Given:
- \( R = 5 \) cm (the radius),
- \( S = 13 \) cm (the slant height).

To find the height \( h \) of the cone, you can use the Pythagorean theorem since \( h \), \( R \), and \( S \) form a right triangle:

\[
S^2 = R^2 + h^2
\]

Substituting the given values:

\[
13^2 = 5^2 + h^2
\]

Calculating:

\[
169 = 25 + h^2
\]

\[
169 - 25 = h^2
\]

\[
144 = h^2
\]

Taking the square root:

\[
h = 12 \text{ cm}
\]

Now, substitute \( R \) and \( h \) back into the volume formula:

\[
V = \frac{1}{3} \pi (5^2)(12)
\]

Calculating:

\[
V = \frac{1}{3} \pi (25)(12)
\]
\[
V = \frac{1}{3} \pi (300)
\]
\[
V = 100\pi
\]

Thus, the volume \( V \) of the cone is approximately:

\[
V \approx 314.16 \text{ cm}^3 \text{ (using } \pi \approx 3.14\text{)}
\]

### Response Insertion Steps:

- **Step 1:** Insert the height \( h = 12 \) cm.
- **Step 2:** Insert the calculated volume \( V = 100\pi \text{ cm}^3 \).
- **Step 3:** If needed, approximate \( V \approx 314.16 \text{ cm}^3 \).

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