5 persons will live in a tent. If each person requires `16 m^(2)` of floor area and `100 m^(3)` space for air then the height of the cone of the smallest size to accommodate these persons would be

Options

A. 16 metre
B. 10.25 metre
C. 20 metre
D. 18.75 metre

Answer

To solve the problem, we need to determine the height of a cone that can accommodate 5 persons, given their space requirements.

### Step 1: Calculate the total floor area required
Each person requires 16 m² of floor area. For 5 persons, the total floor area required is:
\[ \text{Total Floor Area} = \text{Number of Persons} \times \text{Area per Person} = 5 \times 16 = 80 \, m² \]

### Step 2: Calculate the total volume of air required
Each person requires 100 m³ of space for air. For 5 persons, the total volume required is:
\[ \text{Total Volume} = \text{Number of Persons} \times \text{Volume per Person} = 5 \times 100 = 500 \, m³ \]

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

### Step 4: Determine the radius from the floor area
The floor area \( A \) of the cone is given by:
\[ A = \pi r^2 \]
We already calculated the total floor area required as 80 m². Thus, we can set up the equation:
\[ \pi r^2 = 80 \]
From this, we can solve for \( r^2 \):
\[ r^2 = \frac{80}{\pi} \]

### Step 5: Substitute \( r^2 \) into the volume formula
Now we can substitute \( r^2 \) into the volume formula:
\[ 500 = \frac{1}{3} \pi \left(\frac{80}{\pi}\right) h \]
This simplifies to:
\[ 500 = \frac{80h}{3} \]

### Step 6: Solve for \( h \)
To find \( h \), we rearrange the equation:
\[ h = \frac{500 \times 3}{80} = \frac{1500}{80} = 18.75 \, m \]

### Conclusion
The height of the cone of the smallest size to accommodate these persons would be:
\[ \boxed{18.75 \, m} \]