The reverberation period T of a room on its volume V, its surface area A and velocity of sound C. If K is dimensions constant, then T =

Options

A. `(KV)/(CA)`
B. `sqrt((KV)/(CA))`
C. `(KV)/(Csqrt(A))`
D. `K/C sqrt(VA)`

Answer

To solve the problem regarding the reverberation period \( T \) of a room in terms of its volume \( V \), surface area \( A \), and velocity of sound \( C \), we will use dimensional analysis. Here’s the step-by-step solution:

### Step 1: Identify the dimensions of each variable
1. **Reverberation period \( T \)**: The dimension of time is given by \([T]\).
2. **Volume \( V \)**: The dimension of volume is \([L^3]\) (length cubed).
3. **Surface area \( A \)**: The dimension of surface area is \([L^2]\) (length squared).
4. **Velocity of sound \( C \)**: The dimension of velocity is \([L T^{-1}]\) (length per time).

### Step 2: Set up the equation
We assume that the reverberation period \( T \) can be expressed as a function of \( V \), \( A \), and \( C \):
\[ T = K \cdot V^a \cdot A^b \cdot C^d \]
where \( K \) is a dimensionless constant, and \( a \), \( b \), and \( d \) are the powers we need to determine.

### Step 3: Write the dimensions of the right-hand side
Substituting the dimensions into the equation gives:
\[ [T] = [L^3]^a \cdot [L^2]^b \cdot [L T^{-1}]^d \]
This simplifies to:
\[ [T] = [L^{3a + 2b + d} \cdot T^{-d}] \]

### Step 4: Equate the dimensions
Now we equate the dimensions on both sides:
1. For length: \( 3a + 2b + d = 0 \)
2. For time: \( -d = 1 \)

### Step 5: Solve the equations
From the second equation, we find:
\[ d = -1 \]

Substituting \( d = -1 \) into the first equation:
\[ 3a + 2b - 1 = 0 \implies 3a + 2b = 1 \]

### Step 6: Choose values for \( a \) and \( b \)
To solve for \( a \) and \( b \), we can use trial and error. Let's start by assuming \( a = 1 \):
\[ 3(1) + 2b = 1 \implies 3 + 2b = 1 \implies 2b = -2 \implies b = -1 \]

### Step 7: Write the final expression for \( T \)
Now we have:
- \( a = 1 \)
- \( b = -1 \)
- \( d = -1 \)

Substituting these values back into the expression for \( T \):
\[ T = K \cdot V^1 \cdot A^{-1} \cdot C^{-1} = K \cdot \frac{V}{A \cdot C} \]

### Final Answer
Thus, the expression for the reverberation period \( T \) is:
\[ T = K \cdot \frac{V}{A \cdot C} \]