Two sources of sound of the same frequency produce sound intensities `I` and `4I` at a point `P` when used individually. If they are used together such that the sounds from them reach `P` with a phase difference of `2pi//3`, the intensity at `P` will be

Options

A. 2l
B. 3l
C. 4l
D. 5l

Explanation

To solve the problem, we will use the formula for the resultant intensity when two sound waves interfere. The formula is given by:

\[ I = I_1 + I_2 + 2\sqrt{I_1 I_2} \cos \phi \]

where:
- \(I_1\) and \(I_2\) are the intensities of the two sound sources,
- \(\phi\) is the phase difference between the two waves.

### Step 1: Identify the intensities
Given:
- \(I_1 = I\)
- \(I_2 = 4I\)

### Step 2: Identify the phase difference
The phase difference is given as:
- \(\phi = \frac{2\pi}{3}\)

### Step 3: Substitute values into the formula
Now we substitute the values into the intensity formula:

\[ I = I_1 + I_2 + 2\sqrt{I_1 I_2} \cos \phi \]

Substituting \(I_1\) and \(I_2\):

\[ I = I + 4I + 2\sqrt{I \cdot 4I} \cos\left(\frac{2\pi}{3}\right) \]

### Step 4: Simplify the equation
Calculating the terms:

1. \(I + 4I = 5I\)
2. Now, calculate \(2\sqrt{I \cdot 4I}\):

\[ 2\sqrt{I \cdot 4I} = 2\sqrt{4I^2} = 2 \cdot 2I = 4I \]

3. Now, we need to find \(\cos\left(\frac{2\pi}{3}\right)\):

\[ \cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2} \]

### Step 5: Substitute back into the equation
Now substituting back into the intensity equation:

\[ I = 5I + 4I \left(-\frac{1}{2}\right) \]

Calculating \(4I \left(-\frac{1}{2}\right)\):

\[ 4I \left(-\frac{1}{2}\right) = -2I \]

So, we have:

\[ I = 5I - 2I = 3I \]

### Final Result
The intensity at point \(P\) when both sources are used together is:

\[ \boxed{3I} \]