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To solve this problem, we need to find the centroid (\( \bar{x}, \bar{y} \)) of the composite shape shown in the image. The shape consists of a rectangle, a triangle, and a half-circle, and we need to use the given dimensions to calculate the centroid.

### Step 1: Divide the Shape into Simpler Components
The shape can be divided into three basic parts:
1. **Rectangle** with dimensions \( b \times h \).
2. **Triangle** with base \( a \) and height \( c \).
3. **Half-Circle** with radius \( d \).

### Step 2: Calculate the Area and Centroid Coordinates of Each Shape

1. **Rectangle:**
   - Area: \( A_1 = b \times h \).
   - Centroid coordinates: \( (\bar{x}_1, \bar{y}_1) \).
     - \( \bar{x}_1 = \frac{b}{2} \).
     - \( \bar{y}_1 = \frac{h}{2} \).

2. **Triangle:**
   - Area: \( A_2 = \frac{1}{2} \times a \times c \).
   - Centroid coordinates: \( (\bar{x}_2, \bar{y}_2) \).
     - \( \bar{x}_2 = \frac{a}{3} \).
     - \( \bar{y}_2 = \frac{c}{3} \).

3. **Half-Circle:**
   - Area: \( A_3 = \frac{1}{2} \pi d^2 \).
   - Centroid coordinates: \( (\bar{x}_3, \bar{y}_3) \).
     - \( \bar{x}_3 = \frac{4d}{3\pi} \).
     - \( \bar{y}_3 = \frac{d}{2} \).

### Step 3: Compute the Centroid of the Composite Shape
The overall centroid \( (\bar{x}, \bar{y}) \) can be found using the weighted average of the centroids of each shape:

\[
\bar{x} = \frac{\sum A_i \bar{x}_i}{\sum A_i}, \quad \bar{y} = \frac{\sum A_i \bar{y}_i}{\sum A_i}.
\]

### Step 4: Plug in the Given Dimensions
Given values:
- \( a = 46 \, \text{mm} \)
- \( b = 54 \, \text{mm} \)
- \( c = 19 \, \text{mm} \)
- \( d = 23 \, \text{mm} \)
- \( h = 24 \, \text{mm} \)

Let’s compute each component's area and centroid coordinates and then find the overall centroid.

Let's proceed with these calculations.The coordinates of the centroid of the shape are:

- \( \bar{x} = 19.42 \, \text{mm} \)
- \( \bar{y} = 10.87 \, \text{mm} \)

These values represent the location of the centroid of the composite shape.

### Questions to Explore Further:
1. How would the centroid location change if the half-circle were a full circle?
2. What impact does changing the dimensions of the triangle have on the overall centroid?
3. How would adding another geometric shape affect the centroid calculation?
4. What happens to the centroid location if the half-circle is replaced by a quarter-circle?
5. How can you verify these centroid calculations experimentally?

**Tip:** Always check if the centroid makes sense visually in the context of the shape layout; it should be roughly in the “center of mass” of the shape components.

The curved boundary of the shape shown is a half-circle. Given the dimensions above, find: (a) the x location of the centroid of the shape, and (b) the y location of the centroid of the shape.

Learn how to calculate the centroid of a composite shape consisting of a rectangle, triangle, and half-circle. This step-by-step solution explains how to find the x and y coordinates of the centroid using geometric decomposition and weighted averages.

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