Find the volume of a right circular cone that has a height of 17.5 meters and a base with a circumference of 19.4 meters. Round your answer to the nearest tenth cubic meter.

To find the volume of a right circular cone, we use the formula:

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

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

We are given the height \( h = 17.5 \) meters and the circumference \( C = 19.4 \) meters of the base. First, we need to find the radius using the relationship between circumference and radius:

\[
C = 2 \pi r
\]

Solving for \( r \):

\[
r = \frac{C}{2 \pi} = \frac{19.4}{2 \pi}
\]

Now, let's calculate the radius:

\[
r \approx \frac{19.4}{6.2832} \approx 3.088 \, \text{meters}
\]

Next, substitute \( r = 3.088 \) meters and \( h = 17.5 \) meters into the volume formula:

\[
V = \frac{1}{3} \pi (3.088)^2 (17.5)
\]

\[
V \approx \frac{1}{3} \pi (9.538) (17.5)
\]

\[
V \approx \frac{1}{3} \pi (167.915)
\]

\[
V \approx \frac{1}{3} \times 527.519 \approx 175.84 \, \text{cubic meters}
\]

Rounding to the nearest tenth, the volume of the cone is approximately **175.8 cubic meters**.

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you find the surface area of a cone given the radius and height?
2. What happens to the volume if the height of the cone is doubled?
3. How does the volume of a cone compare to the volume of a cylinder with the same base and height?
4. How would you calculate the slant height of a cone given its height and radius?
5. How does changing the radius affect the volume of the cone?

**Tip:** When dealing with cone problems, always check if you need to find the radius from the circumference or diameter, as it's a common intermediate step!

Learn how to calculate the volume of a right circular cone with a height of 17.5 meters and a base circumference of 19.4 meters. This geometry problem demonstrates how to use the volume formula V = (1/3)πr²h and involves finding the radius from the given circumference. Suitable for students in Grades 9-12.

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