Let’s analyze the problem presented in the image. Here's a breakdown:

Part (b): For the two beam sections, you need to determine:

1. Centroidal axes (x-x and y-y):

   • The location of the centroid involves dividing the beam into its constituent rectangular sections and using the formula for the centroidal axis location: $\bar{y} = \frac{\sum A_i y_i}{\sum A_i}, \quad \bar{x} = \frac{\sum A_i x_i}{\sum A_i}$ Where:
     • $A_i$: Area of each section.
     • $x_i, y_i$: Distance of each section's centroid from a reference axis.

2. Neutral axis, $y$:

   • The neutral axis corresponds to the $y$-coordinate of the centroid.

3. Moment of inertia, $I_{xx}$:

   • Use the parallel axis theorem: $I_{xx} = \sum \left(I_{xx,i} + A_i d_i^2 \right)$ Where:
     • $I_{xx,i}$: Moment of inertia of each section about its centroidal axis.
     • $d_i$: Distance from the section's centroidal axis to the overall centroid.

4. Section modulus, $Z_{xx}$:

   • Section modulus is calculated as: $Z_{xx} = \frac{I_{xx}}{\bar{y}_{\text{max}}}$ Where $\bar{y}_{\text{max}}$ is the distance from the centroid to the farthest point in the cross-section.

Part (c): Circular sewer measurement error:

1. Given:

   • The radius is measured 2% too large.
   • This affects the area of the circular cross-section: $A = \pi r^2$
   • An increase in radius by $\delta r$ results in: $\Delta A = 2 \pi r \Delta r + (\pi (\Delta r)^2)$
   • Using the binomial approximation, for small $\Delta r/r$: (1 + x)^n \approx 1 + nx \quad \text{(for small x)}

   Determine the percentage error in the area based on the given overestimation in radius.

Next Steps:

Would you like me to:

1. Solve (b) for the first beam (left diagram)?
2. Solve (b) for the second beam (right diagram)?
3. Tackle part (c) directly?

Let me know! I can also provide more detailed step-by-step explanations for each.