The base area and volume of a conical tent are `154 m^(2)` and `1232 m^(3)`. Find the area of cloth used in this tent.

Answer

To find the area of cloth used in the conical tent, we need to calculate the curved surface area of the cone. Here’s a step-by-step solution:

### Step 1: Understand the given information
- Base area of the cone (A) = 154 m²
- Volume of the cone (V) = 1232 m³

### Step 2: Use the formula for the volume of a cone
The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height.

### Step 3: Use the base area to find the radius
The base area \( A \) of the cone is given by:
\[ A = \pi r^2 \]
We can rearrange this to find \( r^2 \):
\[ r^2 = \frac{A}{\pi} = \frac{154}{\pi} \]
Using \( \pi \approx \frac{22}{7} \):
\[ r^2 = \frac{154 \times 7}{22} = 49 \]
Taking the square root gives us:
\[ r = 7 \text{ m} \]

### Step 4: Substitute the values into the volume formula to find the height
Now substitute \( r \) into the volume formula:
\[ 1232 = \frac{1}{3} \pi (7^2) h \]
\[ 1232 = \frac{1}{3} \pi (49) h \]
Substituting \( \pi \approx \frac{22}{7} \):
\[ 1232 = \frac{1}{3} \times \frac{22}{7} \times 49 \times h \]
Calculating \( \frac{22 \times 49}{21} \):
\[ 1232 = \frac{1078}{3} h \]
Multiplying both sides by 3:
\[ 3696 = 1078h \]
Now, solving for \( h \):
\[ h = \frac{3696}{1078} = 24 \text{ m} \]

### Step 5: Calculate the slant height \( L \)
The slant height \( L \) of the cone can be found using the Pythagorean theorem:
\[ L = \sqrt{r^2 + h^2} \]
Substituting the values we found:
\[ L = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25 \text{ m} \]

### Step 6: Calculate the curved surface area (CSA)
The curved surface area \( CSA \) of the cone is given by:
\[ CSA = \pi r L \]
Substituting the values:
\[ CSA = \pi \times 7 \times 25 \]
Using \( \pi \approx \frac{22}{7} \):
\[ CSA = \frac{22}{7} \times 7 \times 25 = 22 \times 25 = 550 \text{ m}^2 \]

### Final Answer
The area of cloth used in the tent is **550 m²**.

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