An AM transmitter radiates 70 kW of carrier power. The radiated power at 75% modulation would be

Options

A. `89.69` kW
B. 79.20 kW
C. 85.59 kW
D. 83.69 kW

Answer

To solve the problem of finding the radiated power of an AM transmitter at 75% modulation, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Values:**
- Carrier Power (Pc) = 70 kW
- Modulation Index (µ) = 75% = 0.75 (convert percentage to decimal)

2. **Use the Formula for Radiated Power (Pt):**
The formula to calculate the total radiated power (Pt) in an AM transmitter is:
\[ P_t = P_c \left(1 + \frac{\mu^2}{2}\right) \]

3. **Substitute the Values into the Formula:**
\[ P_t = 70 \, \text{kW} \left(1 + \frac{(0.75)^2}{2}\right) \]

4. **Calculate the Modulation Term:**
- First, calculate \((0.75)^2\):
\[ (0.75)^2 = 0.5625 \]
- Now, substitute this value back into the equation:
\[ P_t = 70 \, \text{kW} \left(1 + \frac{0.5625}{2}\right) \]
- Calculate \(\frac{0.5625}{2}\):
\[ \frac{0.5625}{2} = 0.28125 \]
- Now, add this to 1:
\[ 1 + 0.28125 = 1.28125 \]

5. **Final Calculation of Radiated Power:**
\[ P_t = 70 \, \text{kW} \times 1.28125 \]
\[ P_t = 89.6875 \, \text{kW} \]

6. **Round the Result:**
The final value can be rounded to two decimal places:
\[ P_t \approx 89.69 \, \text{kW} \]

### Final Answer:
The radiated power at 75% modulation is approximately **89.69 kW**.