The base radius and total surface area of a cone are 7 cm and `704 cm^(2)`. Find its slant height.

Answer

To find the slant height of the cone, we can use the formula for the total surface area of a cone, which is given by:

\[ \text{Total Surface Area} = \pi r l + \pi r^2 \]

Where:
- \( r \) is the base radius,
- \( l \) is the slant height,
- \( \pi \) is approximately \( \frac{22}{7} \) or \( 3.14 \).

Given:
- Base radius \( r = 7 \, \text{cm} \)
- Total surface area \( = 704 \, \text{cm}^2 \)

### Step-by-Step Solution:

1. **Write the formula for the total surface area of the cone:**
\[ \text{Total Surface Area} = \pi r l + \pi r^2 \]

2. **Substitute the known values into the formula:**
\[ 704 = \pi \cdot 7 \cdot l + \pi \cdot 7^2 \]

3. **Calculate \( \pi \cdot 7^2 \):**
\[ 7^2 = 49 \quad \Rightarrow \quad \pi \cdot 7^2 = \frac{22}{7} \cdot 49 = 22 \cdot 7 = 154 \]

4. **Substitute \( \pi \cdot 7^2 \) back into the equation:**
\[ 704 = \frac{22}{7} \cdot 7 \cdot l + 154 \]

5. **Simplify the equation:**
\[ 704 = 22l + 154 \]

6. **Isolate the term with \( l \):**
\[ 22l = 704 - 154 \]

7. **Calculate \( 704 - 154 \):**
\[ 704 - 154 = 550 \]

8. **Now, solve for \( l \):**
\[ 22l = 550 \quad \Rightarrow \quad l = \frac{550}{22} \]

9. **Calculate \( \frac{550}{22} \):**
\[ l = 25 \]

10. **Conclusion:**
The slant height \( l \) of the cone is \( 25 \, \text{cm} \).

### Final Answer:
The slant height of the cone is \( 25 \, \text{cm} \).

---