The time of reverberation of a room A is one second. What will be the time (in seconds) of reverberation of room, having all the dimensions double of those of room A?

Options

A. `2`
B. `4`
C. `(1)/(2)`
D. `1`

Explanation

To solve the problem of finding the time of reverberation of a room with dimensions double those of room A, we can follow these steps:

### Step 1: Understand the formula for reverberation time
The time of reverberation (T) can be calculated using the formula:
\[ T = 0.16 \frac{V}{A} \]
where:
- \( T \) = time of reverberation
- \( V \) = volume of the room
- \( A \) = total absorption of the room (which depends on the surface area and absorption coefficient)

### Step 2: Determine the volume and surface area of room A
Let the dimensions of room A be \( l, w, h \) (length, width, height). The volume \( V_A \) and surface area \( S_A \) of room A can be calculated as:
\[ V_A = l \times w \times h \]
\[ S_A = 2(lw + lh + wh) \]

### Step 3: Calculate the volume and surface area of the new room
If the dimensions of the new room are doubled, then:
- Length = \( 2l \)
- Width = \( 2w \)
- Height = \( 2h \)

The new volume \( V_B \) and surface area \( S_B \) will be:
\[ V_B = (2l) \times (2w) \times (2h) = 8 \times V_A \]
\[ S_B = 2((2l)(2w) + (2l)(2h) + (2w)(2h)) = 4 \times S_A \]

### Step 4: Set up the ratio of reverberation times
Using the formula for reverberation time, we can express the reverberation time for room B:
\[ T_B = 0.16 \frac{V_B}{A_B} \]
Since the absorption coefficient \( A \) does not change, we can express the ratio of the reverberation times:
\[ \frac{T_A}{T_B} = \frac{V_A / S_A}{V_B / S_B} \]

Substituting the values:
\[ \frac{T_A}{T_B} = \frac{V_A / S_A}{8 \times V_A / (4 \times S_A)} \]
This simplifies to:
\[ \frac{T_A}{T_B} = \frac{1}{2} \]

### Step 5: Calculate the new reverberation time
Given that the time of reverberation for room A (\( T_A \)) is 1 second:
\[ T_B = 2 \times T_A = 2 \times 1 \text{ second} = 2 \text{ seconds} \]

### Final Answer
The time of reverberation for the room with dimensions double those of room A is **2 seconds**.

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