This question asks us to calculate the discharge ($Q$) in a stone sewer using the given parameters: Area ($A$), Wetted Perimeter ($P$), Slope ($S$), and Manning's roughness coefficient ($N$). We will use Manning's formula, a fundamental equation in open channel flow analysis.

Sewer Parameters Identification

First, let's list the known values provided in the question:

• Area of flow, $A = 4$ m$^2$
• Wetted Perimeter, $P = 1$ m
• Slope of the sewer, $S = \frac{1}{100} = 0.01$
• Manning's roughness coefficient, $N = 0.01$

Calculating Hydraulic Radius

Manning's formula requires the hydraulic radius ($R$), which is defined as the ratio of the flow area ($A$) to the wetted perimeter ($P$).

The formula for hydraulic radius is:

$$ R = \frac{A}{P} $$

Substituting the given values:

$$ R = \frac{4 \text{ m}^2}{1 \text{ m}} $$

$$ R = 4 \text{ m} $$

Applying Manning's Formula

Manning's formula for discharge ($Q$) in an open channel is:

$$ Q = \frac{1}{N} A R^{\frac{2}{3}} S^{\frac{1}{2}} $$

Where:

• $Q$ is the discharge in cubic meters per second (cumecs)
• $N$ is Manning's roughness coefficient
• $A$ is the cross-sectional area of flow
• $R$ is the hydraulic radius
• $S$ is the slope of the channel

Discharge Calculation Steps

Now, let's plug the values into Manning's formula:

1. Substitute the known values:

   $$ Q = \frac{1}{0.01} \times 4 \text{ m}^2 \times (4 \text{ m})^{\frac{2}{3}} \times (0.01)^{\frac{1}{2}} $$

2. Simplify the terms:

   $$ Q = 100 \times 4 \times (4)^{\frac{2}{3}} \times 0.1 $$

3. Calculate $(4)^{\frac{2}{3}}$:

   $$ (4)^{\frac{2}{3}} \approx 2.5198 $$

4. Continue the calculation:

   $$ Q = 100 \times 4 \times 2.5198 \times 0.1 $$

   $$ Q = 400 \times 2.5198 \times 0.1 $$

   $$ Q = 1007.92 \times 0.1 $$

   $$ Q \approx 100.792 \text{ cumecs} $$

5. Round the result to one decimal place:

   $$ Q \approx 100.8 \text{ cumecs} $$

Final Answer Determination

Comparing our calculated discharge with the given options, the closest value is 100.8 cumecs.

Therefore, the discharge into the sewer is approximately 100.8 cumecs.