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

Options

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

Explanation

To find the height of the cone of the smallest size to accommodate 5 persons, we will follow these steps:

### Step 1: Calculate the total floor area required for 5 persons.
Each person requires \(16 \, m^2\) of floor area. Therefore, for 5 persons:
\[ \text{Total floor area} = 5 \times 16 = 80 \, m^2 \]

### Step 2: Relate the floor area to the radius of the cone.
The floor area of a cone is given by the formula:
\[ \text{Area} = \pi r^2 \]
Setting this equal to the total floor area we calculated:
\[ \pi r^2 = 80 \]
To find \(r^2\):
\[ r^2 = \frac{80}{\pi} \]

### Step 3: Calculate the total volume of air required for 5 persons.
Each person requires \(100 \, m^3\) of air. Therefore, for 5 persons:
\[ \text{Total volume} = 5 \times 100 = 500 \, m^3 \]

### Step 4: Use the volume formula for a 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 we calculated:
\[ \frac{1}{3} \pi r^2 h = 500 \]

### Step 5: Substitute \(r^2\) into the volume equation.
Substituting \(r^2 = \frac{80}{\pi}\) into the volume equation:
\[ \frac{1}{3} \pi \left(\frac{80}{\pi}\right) h = 500 \]
This simplifies to:
\[ \frac{80}{3} h = 500 \]

### Step 6: Solve for \(h\).
To find \(h\), multiply both sides by 3:
\[ 80h = 1500 \]
Now, divide by 80:
\[ h = \frac{1500}{80} = \frac{75}{4} = 18.75 \, m \]

### Final Answer:
The height of the cone of the smallest size to accommodate the 5 persons is \(18.75 \, m\).

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