The circumference of the base of a cone is `24 pi cm` and its vertical height is 5 cm. Find its slant height.

Answer

To find the slant height of the cone, we will follow these steps:

### Step 1: Identify the given values
- Circumference of the base of the cone = \( 24\pi \) cm
- Vertical height of the cone (h) = 5 cm

### Step 2: Use the formula for the circumference of a circle
The formula for the circumference \( C \) of a circle is given by:
\[ C = 2\pi r \]
Where \( r \) is the radius of the base of the cone.

### Step 3: Set up the equation
From the problem, we have:
\[ 2\pi r = 24\pi \]

### Step 4: Simplify the equation to find the radius
We can divide both sides by \( 2\pi \):
\[ r = \frac{24\pi}{2\pi} = 12 \text{ cm} \]
So, the radius \( r \) of the cone is 12 cm.

### Step 5: Use the Pythagorean theorem to find the slant height
In a right triangle formed by the radius, height, and slant height of the cone, we can apply the Pythagorean theorem:
\[ L^2 = r^2 + h^2 \]
Where:
- \( L \) is the slant height
- \( r \) is the radius (12 cm)
- \( h \) is the height (5 cm)

### Step 6: Substitute the known values into the equation
Substituting the values we have:
\[ L^2 = 12^2 + 5^2 \]
\[ L^2 = 144 + 25 \]
\[ L^2 = 169 \]

### Step 7: Solve for the slant height
To find \( L \), take the square root of both sides:
\[ L = \sqrt{169} = 13 \text{ cm} \]

### Conclusion
The slant height of the cone is \( 13 \) cm.

---