A conical tent is the accommodate 10 persons. Each person must have `6 m^(2)` space to sit and `30 m^(3)` of air to breathe. What will be the height of the cone

Options

A. 37.5 m
B. 150 m
C. 75 m
D. none of these

Answer

To find the height of the conical tent that can accommodate 10 persons, we need to consider both the area required for sitting and the volume of air required for breathing.

### Step 1: Calculate the total area required for sitting
Each person requires 6 m² of space. Therefore, for 10 persons:
\[ \text{Total area} = 10 \times 6 = 60 \, \text{m}^2 \]

### Step 2: Relate the area to the base of the cone
The area of the base of the cone (which is circular) is given by the formula:
\[ \text{Area} = \pi r^2 \]
Setting this equal to the total area required:
\[ \pi r^2 = 60 \]

### Step 3: Solve for \( r^2 \)
Using \( \pi \approx \frac{22}{7} \):
\[ \frac{22}{7} r^2 = 60 \]
Multiplying both sides by \( \frac{7}{22} \):
\[ r^2 = 60 \times \frac{7}{22} = \frac{420}{22} = \frac{210}{11} \]

### Step 4: Calculate the total volume of air required
Each person requires 30 m³ of air. Therefore, for 10 persons:
\[ \text{Total volume} = 10 \times 30 = 300 \, \text{m}^3 \]

### Step 5: Relate the volume to the cone
The volume \( V \) of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
Setting this equal to the total volume required:
\[ \frac{1}{3} \pi r^2 h = 300 \]

### Step 6: Substitute \( r^2 \) into the volume equation
Substituting \( r^2 = \frac{210}{11} \) into the volume equation:
\[ \frac{1}{3} \cdot \frac{22}{7} \cdot \frac{210}{11} \cdot h = 300 \]

### Step 7: Simplify the equation
First, calculate \( \frac{22 \cdot 210}{3 \cdot 7 \cdot 11} \):
\[ \frac{22 \cdot 210}{231} = \frac{4620}{231} = 20 \]
Thus, we have:
\[ 20h = 300 \]

### Step 8: Solve for \( h \)
Dividing both sides by 20:
\[ h = \frac{300}{20} = 15 \, \text{m} \]

### Final Answer
The height of the conical tent is:
\[ \boxed{15 \, \text{m}} \]