Answer:
To find the base radius, height, and volume of the cone, we'll use the given information and the formulas for the properties of a cone.
1. **Given Data:**
- Slant height (\( l \)) = 10.5 cm
- Curved surface area (CSA) = 115.5 cm²
2. **Formulas:**
- Curved Surface Area (\( \text{CSA} \)) = \(\pi r l\)
- Base radius (\( r \)) = ?
- Height (\( h \)) = ?
- Volume (\( V \)) = \(\frac{1}{3} \pi r^2 h\)
3. **Calculate the Base Radius:**
From the formula for the curved surface area:
\[
\text{CSA} = \pi r l
\]
Substitute the known values:
\[
115.5 = \pi r \times 10.5
\]
Solve for \( r \):
\[
r = \frac{115.5}{\pi \times 10.5}
\]
\[
r = \frac{115.5}{32.98672} \approx 3.50 \text{ cm} \quad \text{(to 3 significant figures)}
\]
4. **Calculate the Height:**
Using the Pythagorean theorem for the right triangle formed by the height, radius, and slant height:
\[
l^2 = r^2 + h^2
\]
Substitute the known values:
\[
10.5^2 = 3.50^2 + h^2
\]
\[
110.25 = 12.25 + h^2
\]
\[
h^2 = 110.25 - 12.25
\]
\[
h^2 = 98
\]
\[
h = \sqrt{98} \approx 9.90 \text{ cm} \quad \text{(to 3 significant figures)}
\]
5. **Calculate the Volume:**
Using the volume formula:
\[
V = \frac{1}{3} \pi r^2 h
\]
Substitute the known values:
\[
V = \frac{1}{3} \pi (3.50)^2 (9.90)
\]
\[
V = \frac{1}{3} \pi (12.25) (9.90)
\]
\[
V = \frac{1}{3} \pi (121.275)
\]
\[
V = \frac{1}{3} \times 3.14159 \times 121.275 \approx 127 \text{ cm}^3 \quad \text{(to 3 significant figures)}
\]
**Summary:**
- Base radius (\( r \)) = 3.50 cm (to 3 significant figures)
- Height (\( h \)) = 9.90 cm (to 3 significant figures)
- Volume (\( V \)) = 127 cm³ (to 3 significant figures)