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Let's break down the problem based on the given image.

You are asked to calculate the moment of inertia and centroid of various geometric shapes within the composite cross section, which is a key concept in mechanics.

### Steps to solve:

1. **Moment of inertia of individual shapes**:
   - **a. Triangle** about its own horizontal centroidal axis.
   - **b. Semicircle** about its own horizontal centroidal axis.
   - **c. Circle (hole)** about its own horizontal centroidal axis (negative since it's a hole).

2. **Composite centroid**:
   - **e. The centroidal position in the y-direction** for the entire composite cross-section.

3. **Moment of inertia of the composite section**:
   - **f. Overall moment of inertia** of the entire cross-section about the composite centroidal axis.

#### Formulas to Use:

1. **Moment of inertia for a triangle** (base \( b \), height \( h \)):
   \[
   I_{\text{triangle}} = \frac{b h^3}{36}
   \]

2. **Moment of inertia for a semicircle** (radius \( r \)):
   \[
   I_{\text{semicircle}} = 0.11 r^4
   \]

3. **Moment of inertia for a circle (hole)**:
   \[
   I_{\text{circle}} = \frac{\pi r^4}{4}
   \]

4. **Centroid formula for composite shapes**:
   \[
   \bar{y} = \frac{\sum (A_i y_i)}{\sum A_i}
   \]
   where \( A_i \) is the area of each shape and \( y_i \) is the centroid of each shape from a reference line.

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Would you like to go step-by-step into solving each part (a through f) using the given dimensions?

Also, here are 5 related questions to expand on this topic:
1. How do you calculate the centroid of composite shapes?
2. What is the difference between the parallel axis theorem and centroidal moment of inertia?
3. How do the moments of inertia for different shapes compare in terms of magnitude?
4. How does the inclusion of a hole affect the moment of inertia of a structure?
5. Why is moment of inertia important in structural engineering?

**Tip**: Always ensure you align each shape's centroid with the reference axis to avoid miscalculations.

For the cross section (Engineering Mechanics, Static by Pytel & Kiusalaas, 3rd edition), determine: a. The moment of inertia of the triangle about its own horizontal centroidal axis. b. The moment of inertia of the semicircle about its own horizontal centroidal axis. c. The moment of inertia of the circle (hole) about its own horizontal centroidal axis. e. The y-direction centroid of the composite cross section. f. The moment of inertia of the composite cross section about its own horizontal centroidal axis.

Calculate the moment of inertia and y-direction centroid for a composite cross section composed of a triangle, a semicircle, and a circular hole. Learn about the key formulas and theorems for calculating moments of inertia and centroids of composite shapes. Suitable for undergraduate engineering students.

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