The slant height of a frustum of a cone is 5cm. If the radius of one end is
2cm and the area of the curved surface is 60cm2 , what is the radius of
the other end?
Find the volume of the frustum.

To solve the problem, we'll follow these steps:

### Step 1: Find the radius of the other end using the formula for the curved surface area of a frustum of a cone.

The formula for the curved surface area (CSA) of a frustum is given by:
\[
\text{CSA} = \pi (r_1 + r_2) l
\]
where:
- \( r_1 \) and \( r_2 \) are the radii of the two ends,
- \( l \) is the slant height.

Given:
- \( l = 5 \) cm,
- \( r_1 = 2 \) cm,
- \(\text{CSA} = 60 \) cm².

We can plug in these values to find \( r_2 \):

\[
60 = \pi (2 + r_2) \times 5
\]
\[
60 = 5\pi (2 + r_2)
\]
\[
\frac{60}{5\pi} = 2 + r_2
\]
\[
\frac{12}{\pi} = 2 + r_2
\]
\[
r_2 = \frac{12}{\pi} - 2
\]

Approximating \( \pi \) as 3.14:

\[
r_2 \approx \frac{12}{3.14} - 2
\]
\[
r_2 \approx 3.82 - 2
\]
\[
r_2 \approx 1.82 \text{ cm}
\]

### Step 2: Find the volume of the frustum.

The formula for the volume \( V \) of a frustum is:
\[
V = \frac{1}{3} \pi h (r_1^2 + r_2^2 + r_1r_2)
\]
where \( h \) is the vertical height. 

First, we need to find the vertical height \( h \) using the Pythagorean theorem:

\[
h = \sqrt{l^2 - (r_2 - r_1)^2}
\]
Substituting the known values:
\[
h = \sqrt{5^2 - (1.82 - 2)^2}
\]
\[
h = \sqrt{25 - (-0.18)^2}
\]
\[
h = \sqrt{25 - 0.0324}
\]
\[
h = \sqrt{24.9676} \approx 4.997 \text{ cm}
\]

Now, substitute the values into the volume formula:
\[
V = \frac{1}{3} \pi \times 4.997 \times \left(2^2 + 1.82^2 + 2 \times 1.82\right)
\]
\[
V = \frac{1}{3} \pi \times 4.997 \times \left(4 + 3.3124 + 3.64\right)
\]
\[
V = \frac{1}{3} \pi \times 4.997 \times 10.9524
\]
\[
V \approx \frac{1}{3} \times 3.14 \times 4.997 \times 10.9524
\]
\[
V \approx 57.41 \text{ cm}^3
\]

### Summary of Results:
- **Radius of the other end \( r_2 \)**: \( \approx 1.82 \) cm
- **Volume of the frustum**: \( \approx 57.41 \) cm³

Would you like more details or have any questions?

---

Here are five related questions to expand your understanding:
1. How would the volume of the frustum change if the slant height were doubled?
2. What is the formula for the total surface area of a frustum?
3. How does the formula for the volume of a frustum relate to that of a full cone?
4. What happens to the volume of the frustum if the radii of both ends are equal?
5. Can the frustum's height ever be greater than its slant height? Why or why not?

**Tip:** Always double-check the dimensions you are working with to avoid confusion between slant height and vertical height.

Discover how to solve the problem of finding the radius of the other end of a frustum of a cone with a given slant height, radius, and curved surface area. Includes detailed steps and calculations for finding both the radius and volume of the frustum. Suitable for Grades 9-12.

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