The base area and height of a right circular conical tent are `154 m^(2)` and 4 m respectively. Find the area of cloth used in the tent.

Answer

To find the area of cloth used in a right circular conical tent, we follow these steps:

### Step 1: Write down the given information
- Base area of the tent (A) = 154 m²
- Height of the tent (h) = 4 m

### Step 2: Find the radius of the base
The area of the base of a cone is given by the formula:
\[ A = \pi r^2 \]
Where:
- \( A \) is the area of the base
- \( r \) is the radius

From the problem, we know:
\[ \pi r^2 = 154 \]

Using \( \pi = \frac{22}{7} \):
\[ \frac{22}{7} r^2 = 154 \]

To find \( r^2 \), we multiply both sides by \( \frac{7}{22} \):
\[ r^2 = 154 \times \frac{7}{22} \]
\[ r^2 = 154 \times \frac{7}{22} = 49 \]

Now, taking the square root:
\[ r = \sqrt{49} = 7 \text{ m} \]

### Step 3: Find the slant height (l)
To find the slant height \( l \) of the cone, we use the Pythagorean theorem:
\[ l^2 = r^2 + h^2 \]
Where:
- \( h \) is the height of the cone

Substituting the known values:
\[ l^2 = 7^2 + 4^2 \]
\[ l^2 = 49 + 16 \]
\[ l^2 = 65 \]

Now, taking the square root:
\[ l = \sqrt{65} \]

### Step 4: Calculate the curved surface area (CSA)
The curved surface area (CSA) of a cone is given by the formula:
\[ \text{CSA} = \pi r l \]

Substituting the values we have:
\[ \text{CSA} = \frac{22}{7} \times 7 \times \sqrt{65} \]

The \( 7 \) in the numerator and denominator cancels out:
\[ \text{CSA} = 22 \sqrt{65} \text{ m}^2 \]

### Final Answer
The area of cloth used in the tent is:
\[ 22 \sqrt{65} \text{ m}^2 \]

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