The circumference of the base of a 12 m high conical tent is 66 m. Find the volume of the air contained in it.

Answer

To find the volume of the air contained in the conical tent, we will follow these steps:

### Step 1: Find the radius of the base of the cone.
The circumference of the base of the cone is given by the formula:
\[ C = 2\pi r \]
where \(C\) is the circumference and \(r\) is the radius.

Given that the circumference \(C = 66\) m, we can set up the equation:
\[ 2\pi r = 66 \]

### Step 2: Solve for \(r\).
To find \(r\), we can rearrange the equation:
\[ r = \frac{66}{2\pi} \]
Using \(\pi \approx \frac{22}{7}\), we substitute:
\[ r = \frac{66}{2 \times \frac{22}{7}} = \frac{66 \times 7}{44} = \frac{462}{44} = \frac{21}{2} = 10.5 \text{ m} \]

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

Given that the height \(h = 12\) m and \(r = 10.5\) m, we can substitute these values into the volume formula:
\[ V = \frac{1}{3} \pi (10.5)^2 (12) \]

### Step 4: Calculate \(r^2\).
First, calculate \(r^2\):
\[ (10.5)^2 = 110.25 \]

### Step 5: Substitute \(r^2\) and calculate the volume.
Now substitute \(r^2\) into the volume formula:
\[ V = \frac{1}{3} \pi (110.25) (12) \]
\[ V = \frac{1}{3} \times \frac{22}{7} \times 110.25 \times 12 \]

### Step 6: Simplify the volume calculation.
Calculating further:
\[ V = \frac{1}{3} \times \frac{22 \times 110.25 \times 12}{7} \]
Calculating \(22 \times 110.25 = 2425.5\):
\[ V = \frac{1}{3} \times \frac{2425.5 \times 12}{7} \]
Calculating \(2425.5 \times 12 = 29106\):
\[ V = \frac{1}{3} \times \frac{29106}{7} \]
Calculating \(\frac{29106}{7} = 4158\):
\[ V = \frac{4158}{3} = 1386 \text{ m}^3 \]

### Final Answer:
The volume of the air contained in the conical tent is \(1386 \text{ m}^3\).

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