What is the peak voltage (in V) of the modulating signal that modulates the message signal wave of amplitude 20 V in order to have a modulation index of 75%?

Explanation

To find the peak voltage of the modulating signal that modulates a message signal wave of amplitude 20 V with a modulation index of 75%, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Modulation Index Formula**:
The modulation index (μ) is defined as the ratio of the amplitude of the modulating signal (AM) to the amplitude of the carrier wave (AC). Mathematically, this is expressed as:
\[ \mu = \frac{A_M}{A_C} \]

2. **Identify Given Values**:
- Amplitude of the carrier wave (AC) = 20 V
- Modulation index (μ) = 75% = 0.75 (as a decimal)

3. **Rearrange the Formula to Find AM**:
We can rearrange the modulation index formula to solve for the amplitude of the modulating signal (AM):
\[ A_M = \mu \times A_C \]

4. **Substitute the Known Values**:
Now, substitute the known values into the rearranged formula:
\[ A_M = 0.75 \times 20 \text{ V} \]

5. **Calculate AM**:
Perform the multiplication:
\[ A_M = 15 \text{ V} \]

### Final Answer:
The peak voltage of the modulating signal is **15 V**.