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The problem involves a right circular cone, with a given height \( h = 15 \) cm and a total surface area of \( 250\pi \, \text{cm}^2 \). We are asked to find the radius \( r \) of the base.

The formula for the total surface area \( A \) of a cone is:
\[
A = \pi r (r + l)
\]
where:
- \( r \) is the radius,
- \( l \) is the slant height.

First, we need to calculate the slant height \( l \). The relationship between the slant height, the height \( h \), and the radius \( r \) is given by the Pythagorean theorem:
\[
l = \sqrt{r^2 + h^2}
\]
Substituting the height \( h = 15 \) cm:
\[
l = \sqrt{r^2 + 15^2} = \sqrt{r^2 + 225}
\]

Now, substituting this expression for \( l \) into the surface area formula:
\[
250\pi = \pi r (r + \sqrt{r^2 + 225})
\]
We can cancel \( \pi \) from both sides:
\[
250 = r (r + \sqrt{r^2 + 225})
\]

Now we solve this equation for \( r \). Let me calculate the solution.The radius \( r \) of the cone is:

\[
r = \frac{50\sqrt{29}}{29} \approx 9.13 \, \text{cm}
\]

Would you like further clarification or details on this solution?

Here are some related questions for further exploration:

1. How do you calculate the volume of this cone?
2. What would the radius be if the height was different but the surface area stayed the same?
3. Can you derive the total surface area formula for a cone?
4. How does changing the radius affect the surface area of a cone?
5. What if only the lateral surface area was given, how would you calculate the radius?

**Tip:** When solving geometric problems involving cones, it's useful to remember the relationship between the slant height, height, and radius using the Pythagorean theorem.

Un cono de revolución tiene una altura de 15 cm y un área total de 250π cm². ¿Cuál es su radio?

Learn how to find the radius of a cone given its height of 15 cm and a total surface area of 250π cm². This solution involves using the surface area formula for a cone and applying the Pythagorean theorem to calculate the slant height. Suitable for students in Grades 9-12.

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