The curved surface are and the slant height of a right circular cone are `858 cm^(2)`. and 13 cm. respectively. Find its diameter (in cm.)

Options

A. 21
B. 42
C. 84
D. 63

Answer

To solve the problem, we need to find the diameter of a right circular cone given its curved surface area (CSA) and slant height (L).

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Curved Surface Area (CSA) = 858 cm²
- Slant Height (L) = 13 cm

2. **Use the Formula for Curved Surface Area of a Cone:**
The formula for the curved surface area of a cone is:
\[ \text{CSA} = \pi R L \]
where \( R \) is the radius of the base of the cone and \( L \) is the slant height.

3. **Substitute the Known Values:**
We can substitute the values we have into the formula. Using \( \pi \approx \frac{22}{7} \):
\[ 858 = \frac{22}{7} R \times 13 \]

4. **Simplify the Equation:**
First, multiply \( \frac{22}{7} \) and \( 13 \):
\[ 858 = \frac{286}{7} R \]

5. **Multiply Both Sides by 7 to Eliminate the Fraction:**
\[ 858 \times 7 = 286 R \]
\[ 6006 = 286 R \]

6. **Solve for R:**
Now, divide both sides by 286:
\[ R = \frac{6006}{286} \]
\[ R = 21 \, \text{cm} \]

7. **Find the Diameter:**
The diameter \( D \) is twice the radius:
\[ D = 2R = 2 \times 21 = 42 \, \text{cm} \]

### Final Answer:
The diameter of the cone is **42 cm**.