The total surface area of a cone is 704 `cm^(2)` and the radius of its base is 7 cm . 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:

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

where:
- \( r \) is the radius of the base,
- \( l \) is the slant height,
- \( \pi \) is a constant approximately equal to 3.14.

Given:
- Total Surface Area = 704 cm²
- Radius \( r = 7 \) cm

We need to find the slant height \( l \).

### Step 1: Substitute the known values into the formula

Substituting the known values into the total surface area formula:

\[ 704 = \pi (7^2) + \pi (7) l \]

### Step 2: Calculate \( \pi (7^2) \)

Calculating \( 7^2 \):

\[ 7^2 = 49 \]

Now substituting this value back into the equation:

\[ 704 = \pi (49) + \pi (7) l \]

### Step 3: Factor out \( \pi \)

Factoring out \( \pi \):

\[ 704 = \pi (49 + 7l) \]

### Step 4: Divide both sides by \( \pi \)

To isolate \( (49 + 7l) \), we divide both sides by \( \pi \):

\[ \frac{704}{\pi} = 49 + 7l \]

Using \( \pi \approx 22/7 \) for calculation:

\[ \frac{704 \times 7}{22} = 49 + 7l \]

Calculating \( \frac{704 \times 7}{22} \):

\[ \frac{704 \times 7}{22} = 224 \]

So now we have:

\[ 224 = 49 + 7l \]

### Step 5: Solve for \( 7l \)

Subtracting 49 from both sides:

\[ 224 - 49 = 7l \]

Calculating \( 224 - 49 \):

\[ 175 = 7l \]

### Step 6: Divide by 7 to find \( l \)

Now, divide both sides by 7:

\[ l = \frac{175}{7} \]

Calculating \( \frac{175}{7} \):

\[ l = 25 \]

### Conclusion

The slant height \( l \) of the cone is:

\[ \text{Slant Height} = 25 \text{ cm} \]