The cloth used in a conical tent is 330 square metre. Find its vertical height if its slant height is 15 m.

Answer

To find the vertical height of the conical tent given the cloth surface area and the slant height, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We know that the cloth used in the conical tent represents the curved surface area of the cone. The given values are:
- Curved Surface Area (CSA) = 330 m²
- Slant Height (l) = 15 m

2. **Formula for Curved Surface Area**:
The formula for the curved surface area of a cone is:
\[ \text{CSA} = \pi r l \]
where \( r \) is the radius and \( l \) is the slant height.

3. **Substituting Known Values**:
We can substitute the known values into the formula:
\[ 330 = \frac{22}{7} \times r \times 15 \]

4. **Solving for Radius (r)**:
Rearranging the equation to solve for \( r \):
\[ r = \frac{330 \times 7}{22 \times 15} \]
Simplifying this:
\[ r = \frac{2310}{330} = 7 \text{ m} \]

5. **Finding the Vertical Height (h)**:
To find the vertical height \( h \) of the cone, we use the Pythagorean theorem:
\[ l^2 = r^2 + h^2 \]
Rearranging gives:
\[ h = \sqrt{l^2 - r^2} \]
Substituting the known values:
\[ h = \sqrt{15^2 - 7^2} \]
Calculating:
\[ h = \sqrt{225 - 49} = \sqrt{176} \]

6. **Simplifying the Height**:
We can simplify \( \sqrt{176} \):
\[ \sqrt{176} = \sqrt{16 \times 11} = 4\sqrt{11} \]

### Final Answer:
The vertical height of the conical tent is \( 4\sqrt{11} \) meters.