An audio signal represented as `25 sin 2pi ( 2000 t)` amplitude modulated by a carrier wave : `60 sin 2pi (100,000)t`. The modulation index of the modulated signal is

Options

A. `25 % `
B. `41.6 % `
C. `50 % `
D. `75%`

Explanation

To find the modulation index of the given amplitude modulated signal, we can follow these steps:

### Step 1: Identify the Amplitudes
The audio signal is given as:
\[ 25 \sin(2\pi(2000)t) \]
The amplitude of the audio signal (message signal) is:
\[ A_m = 25 \]

The carrier wave is given as:
\[ 60 \sin(2\pi(100,000)t) \]
The amplitude of the carrier wave is:
\[ A_c = 60 \]

### Step 2: Use the Formula for Modulation Index
The modulation index \( m \) for amplitude modulation is defined as the ratio of the amplitude of the message signal to the amplitude of the carrier signal:
\[ m = \frac{A_m}{A_c} \]

### Step 3: Substitute the Values
Substituting the values of \( A_m \) and \( A_c \) into the formula:
\[ m = \frac{25}{60} \]

### Step 4: Calculate the Modulation Index
Now, we perform the division:
\[ m = \frac{25}{60} \approx 0.4167 \]

### Step 5: Convert to Percentage (if required)
To express the modulation index as a percentage, we multiply by 100:
\[ m \times 100 = 0.4167 \times 100 \approx 41.67\% \]

### Final Answer
Thus, the modulation index of the modulated signal is approximately:
\[ 41.67\% \]

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