Given data:
Diameter of the pipe (D) = 10 cm = 0.1 m
Velocity of flow (V) = 0.2 m/s
Kinematic viscosity (ν) = 10^-5 m^2/s

Step 1: Reynolds Number Calculation
Reynolds number (Re) is a dimensionless quantity that determines the type of flow in the pipe. It is given by the formula:
Re = (V * D) / ν

Substituting the given values:
Re = (0.2 * 0.1) / 10^-5
Re = 2000

Step 2: Determine the Flow Regime
Based on the Reynolds number, we can determine the flow regime. In this case, the flow is considered to be fully developed, which means it is in the smooth regime.

Step 3: Calculation of Darcy Friction Factor (f)
The Darcy friction factor (f) is a measure of the resistance to flow in a pipe. It can be calculated using the Colebrook equation:
1 / √f = -2 * log10((ε/D)/3.7 + 2.51 / (Re * √f))

Where:
ε is the roughness height of the pipe

Since the flow is considered to be in the smooth regime, we can assume a negligible roughness height (ε) for the pipe. Therefore, the equation simplifies to:
1 / √f = -2 * log10(2.51 / (Re * √f))

Step 4: Solving for Darcy Friction Factor
To solve for f, we need to solve the above equation iteratively. We can start with an assumed value for f and then use the equation to update the value until it converges to a solution.

Using an initial guess of f = 0.03, the equation becomes:
1 / √0.03 = -2 * log10(2.51 / (2000 * √0.03))
1 / √0.03 = -2 * log10(2.51 / (2000 * 0.1732))
1 / √0.03 = -2 * log10(2.51 / 346.4)
1 / √0.03 = -2 * log10(0.007239)
1 / √0.03 = -2 * (-2.14)
1 / √0.03 = 4.28

Taking the reciprocal of both sides:
√0.03 = 1/4.28
0.1732 = 0.2336

The calculated value of √f is 0.2336. Taking the square of both sides gives us the value of f:
f = 0.0544

The calculated value of f is 0.0544, which is not within the given range of 0.03 to 0.035.

Therefore, the correct answer cannot be determined with the given data.