75 persons can sleep in a room 25m by 9.6m. If each person requires `16m^(3)` of air, find the height of the room.

Answer

To find the height of the room, we can follow these steps:

### Step 1: Calculate the total volume of air required for 75 persons.
Each person requires 16 m³ of air. Therefore, for 75 persons, the total volume of air required is:
\[ \text{Total Volume} = \text{Number of Persons} \times \text{Volume per Person} = 75 \times 16 \, \text{m}^3 \]
\[ \text{Total Volume} = 1200 \, \text{m}^3 \]

### Step 2: Write the formula for the volume of the room.
The volume \( V \) of a cuboidal room can be calculated using the formula:
\[ V = \text{Length} \times \text{Breadth} \times \text{Height} \]
Given that the length of the room is 25 m and the breadth is 9.6 m, we can express the volume as:
\[ V = 25 \, \text{m} \times 9.6 \, \text{m} \times h \]

### Step 3: Set the volume of the room equal to the total volume of air required.
Since the total volume of air required is equal to the volume of the room, we can set up the equation:
\[ 25 \times 9.6 \times h = 1200 \]

### Step 4: Calculate the area of the base of the room.
First, calculate the area of the base:
\[ \text{Area} = 25 \times 9.6 = 240 \, \text{m}^2 \]

### Step 5: Solve for the height \( h \).
Now we can substitute the area into the volume equation:
\[ 240 \times h = 1200 \]
To find \( h \), divide both sides by 240:
\[ h = \frac{1200}{240} = 5 \, \text{m} \]

### Conclusion:
The height of the room is \( 5 \, \text{m} \).

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