A transmitter radiates 11.8 kW of power with the carrier unmodulated and 10 kW with the carrier sinusoidally modulated. The modulation factor is

Options

A. `56%`
B. `60%`
C. `72%`
D. `84%`

Explanation

To find the modulation factor (μ) for the given problem, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Power radiated with unmodulated carrier, \( P_t = 11.8 \, \text{kW} \)
- Power radiated with sinusoidally modulated carrier, \( P_c = 10 \, \text{kW} \)

2. **Use the Power Relation:**
The relationship between the power of the carrier and the transmitted power is given by:
\[ P_c = \frac{P_t}{1 + \frac{\mu^2}{2}} \]

3. **Substitute the Known Values:**
Substitute \( P_t \) and \( P_c \) into the equation:
\[ 10 = \frac{11.8}{1 + \frac{\mu^2}{2}} \]

4. **Cross-Multiply to Eliminate the Fraction:**
Rearranging gives:
\[ 10 \left(1 + \frac{\mu^2}{2}\right) = 11.8 \]
Expanding this:
\[ 10 + 5\mu^2 = 11.8 \]

5. **Isolate the Modulation Factor Term:**
Subtract 10 from both sides:
\[ 5\mu^2 = 11.8 - 10 \]
\[ 5\mu^2 = 1.8 \]

6. **Solve for \( \mu^2 \):**
Divide both sides by 5:
\[ \mu^2 = \frac{1.8}{5} = 0.36 \]

7. **Calculate \( \mu \):**
Take the square root of both sides:
\[ \mu = \sqrt{0.36} = 0.6 \]

8. **Convert to Percentage:**
To express \( \mu \) as a percentage:
\[ \mu = 0.6 \times 100\% = 60\% \]

### Final Answer:
The modulation factor \( \mu \) is **60%**.