Given a unity feedback control system with the transfer function G(s) = K/s(s+4), we need to find the value of K for which the damping ratio is 0.5.

Damping Ratio and Transfer Function
The damping ratio is a measure of how fast the system responds to changes and how quickly it settles. It is denoted by the Greek letter ζ (zeta). The transfer function of a system relates the output of the system to its input and is represented by G(s).

Damping Ratio Formula
The damping ratio ζ can be calculated using the following formula:
ζ = ζ1/ζ2
Where ζ1 is the real part of the complex pole and ζ2 is the imaginary part of the complex pole.

Transfer Function in Standard Form
The transfer function G(s) = K/s(s+4) can be rewritten in standard form as:
G(s) = K/(s^2 + 4s)

Calculating the Damping Ratio
To calculate the damping ratio, we need to find the poles of the transfer function. The poles are the values of s for which the denominator of the transfer function becomes zero.
Setting the denominator equal to zero, we get:
s^2 + 4s = 0
s(s + 4) = 0
s = 0 or s = -4

Calculating ζ1 and ζ2:
For s = 0:
ζ1 = 0
ζ2 = 0

For s = -4:
ζ1 = -4
ζ2 = 0

Using the formula for damping ratio:
ζ = ζ1/ζ2
ζ = -4/0 (division by zero is undefined)

Since the denominator is zero, we cannot calculate the damping ratio.

Finding the Value of K
Given that the damping ratio is 0.5, we can conclude that there is a mistake in the options provided. It is not possible to find a value of K for which the damping ratio is 0.5. The correct answer cannot be determined from the options given.

In conclusion, the options provided do not represent the correct answer. The correct answer cannot be determined based on the given information.