The total surface area of a right circular cone of slant height 13 cm is `90pi cm^(2).` Calculate its radius in cm.

Answer

To solve the problem of finding the radius of a right circular cone given its slant height and total surface area, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Slant height (L) = 13 cm
- Total Surface Area (TSA) = 90π cm²

2. **Write the Formula for Total Surface Area of a Cone:**
The formula for the total surface area (TSA) of a cone is:
\[ \text{TSA} = \pi r l + \pi r^2 \]
where \( r \) is the radius and \( l \) is the slant height.

3. **Substitute the Given Values into the Formula:**
Substitute \( l = 13 \) cm and TSA = \( 90\pi \) cm² into the formula:
\[ 90\pi = \pi r (13 + r) \]

4. **Cancel π from Both Sides:**
Since π is a common factor, we can divide both sides by π:
\[ 90 = r(13 + r) \]

5. **Rearrange the Equation:**
Expand the right side:
\[ 90 = 13r + r^2 \]
Rearranging gives:
\[ r^2 + 13r - 90 = 0 \]

6. **Factor the Quadratic Equation:**
We need to factor the quadratic equation \( r^2 + 13r - 90 = 0 \). We look for two numbers that multiply to -90 and add to 13. The numbers are 18 and -5:
\[ (r + 18)(r - 5) = 0 \]

7. **Set Each Factor to Zero:**
From the factored form, we have:
\[ r + 18 = 0 \quad \text{or} \quad r - 5 = 0 \]

8. **Solve for r:**
Solving these gives:
- \( r + 18 = 0 \) → \( r = -18 \) (not possible since radius cannot be negative)
- \( r - 5 = 0 \) → \( r = 5 \) (valid solution)

9. **Conclusion:**
The radius of the cone is:
\[ \text{Radius} = 5 \text{ cm} \]

### Final Answer:
Hence, the required radius is **5 cm**.

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