Explanation:

To understand why the peak transmit power of the radar has to be increased by a factor of 16 when the range is doubled, we need to consider the basic principles of radar operation.

Radar Operation:
Radar works by transmitting a high-frequency electromagnetic wave, known as a radar pulse, and then detecting the echo reflected back from a target. The time taken for the pulse to travel to the target and return is used to calculate the range or distance to the target.

Radar Range Equation:
The range or distance to a target can be calculated using the radar range equation:

R = (c * Δt) / 2

Where:
R = Range to the target
c = Speed of light
Δt = Time taken for the pulse to travel to the target and return

Relationship between Transmit Power and Range:
In the radar range equation, the time taken for the pulse to travel to the target and return (Δt) is directly proportional to the range (R). This means that to increase the range, the time taken for the pulse to travel to the target and return must be increased.

The time taken for the pulse to travel to the target and return is determined by the speed of light (c) and the distance traveled by the pulse. Since the speed of light is constant, the distance traveled by the pulse needs to be increased to increase the time taken.

To increase the distance traveled by the pulse, the transmit power of the radar needs to be increased. This is because the power of the radar pulse decreases with distance due to spreading and attenuation effects.

Relationship between Transmit Power and Range:
The power of a radar pulse decreases with distance according to the inverse square law:

P = P0 / (4πR²)

Where:
P = Power of the radar pulse at a distance R
P0 = Power of the radar pulse at a reference distance

From the above equation, it can be observed that the power of the radar pulse is inversely proportional to the square of the range. Therefore, to double the range, the power of the radar pulse needs to be increased by a factor of 2² = 4.

However, we also need to consider that the transmit power is squared in the radar range equation:

R = (c * Δt) / 2

Therefore, to double the range, the transmit power needs to be increased by a factor of 4² = 16.

Hence, the correct answer is option 'D' - increased by a factor of 16.