The total surface area of a close cone is 176 `cm^(2)`. If its radius is 4 cm, find its slant height.

Answer

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

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

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

Given:
- Total Surface Area = 176 cm²
- Radius \( r = 4 \) cm

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

### Step 2: Substitute the known values into the formula.
\[ 176 = \pi (4) l + \pi (4)^2 \]

### Step 3: Calculate \( \pi (4)^2 \).
\[ \pi (4)^2 = \pi \cdot 16 = 16\pi \]

### Step 4: Substitute \( 16\pi \) back into the equation.
\[ 176 = \pi (4) l + 16\pi \]

### Step 5: Factor out \( \pi \) from the right side.
\[ 176 = \pi (4l + 16) \]

### Step 6: Substitute \( \pi \) with \( \frac{22}{7} \).
\[ 176 = \frac{22}{7} (4l + 16) \]

### Step 7: Multiply both sides by \( 7 \) to eliminate the fraction.
\[ 176 \times 7 = 22(4l + 16) \]

### Step 8: Calculate \( 176 \times 7 \).
\[ 1232 = 22(4l + 16) \]

### Step 9: Divide both sides by \( 22 \).
\[ \frac{1232}{22} = 4l + 16 \]

### Step 10: Calculate \( \frac{1232}{22} \).
\[ 56 = 4l + 16 \]

### Step 11: Subtract \( 16 \) from both sides.
\[ 56 - 16 = 4l \]
\[ 40 = 4l \]

### Step 12: Divide both sides by \( 4 \) to solve for \( l \).
\[ l = \frac{40}{4} = 10 \text{ cm} \]

### Final Answer:
The slant height \( l \) of the cone is \( 10 \) cm.

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