Determine the coordinates of the centroid of the shaded area. y
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![### Centroid Calculation of a Shaded Area
**Objective:**
Determine the coordinates of the centroid of the shaded area shown in the figure.
---
**Diagram Explanation:**
The diagram consists of a composite geometric shape with various dimensions provided in millimeters. The composite shape can be broken down into simpler geometric shapes for the purpose of centroid calculation.
**Dimensions:**
- Total length along the x-axis: 165 mm
- Height along the y-axis: 80 mm
- Distance from the y-axis to the start of the rectangle: 45 mm
- Width of the rectangle: 25 mm
- Distance between the rectangles: 55 mm
- Semi-circle diameter: 65 mm
- Inner distance of the semi-circle: 55 mm
The entire figure is composed of:
1. A trapezoid on the left
2. A rectangle in the central portion
3. A semi-circle on the right
To identify individual shapes and better calculate the centroid, each segment can be approached separately, correctly associating their dimensions to place the center of mass (centroid) accurately:
1. **Trapezoid:**
- Coordinates: Height = 80 mm, Bases = 45 mm and 25 mm, Other side = 65 mm
2. **Rectangle:**
- Coordinates: W = 55 mm, H = 80 mm
3. **Semi-circle:**
- Diameter = 65 mm (radius = 32.5 mm)
Additionally, the trapezoid and rectangle are placed on a baseline where these dimensions become critical for calculating their centroid location.
**Centroid Calculation:**
The centroid \(\bar{x}\) and \(\bar{y}\) can be computed using the formula for composite areas, taking the moments of individual areas about the baseline (x, y reference axes):
1. Break down the composite area into distinct, simpler shapes (trapezoid, rectangle, and semi-circle).
2. Calculate the area (A) and centroid (\(x\), \(y\)) of each simpler shape.
3. Use the formula:
\[ \bar{x} = \frac{\Sigma(A_i \cdot x_i)}{\Sigma A_i} \]
\[ \bar{y} = \frac{\Sigma(A_i \cdot y_i)}{\Sigma A_i} \]
Where:
- \(A_i\) = Area of the ith shape
- \(x](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc59a6d07-a993-40f7-b57e-cdb2ad079d78%2F0b3b11a5-9516-43f0-9693-a241d92f6ce5%2Ftpk1bgc_processed.png&w=3840&q=75)
Transcribed Image Text:### Centroid Calculation of a Shaded Area
**Objective:**
Determine the coordinates of the centroid of the shaded area shown in the figure.
---
**Diagram Explanation:**
The diagram consists of a composite geometric shape with various dimensions provided in millimeters. The composite shape can be broken down into simpler geometric shapes for the purpose of centroid calculation.
**Dimensions:**
- Total length along the x-axis: 165 mm
- Height along the y-axis: 80 mm
- Distance from the y-axis to the start of the rectangle: 45 mm
- Width of the rectangle: 25 mm
- Distance between the rectangles: 55 mm
- Semi-circle diameter: 65 mm
- Inner distance of the semi-circle: 55 mm
The entire figure is composed of:
1. A trapezoid on the left
2. A rectangle in the central portion
3. A semi-circle on the right
To identify individual shapes and better calculate the centroid, each segment can be approached separately, correctly associating their dimensions to place the center of mass (centroid) accurately:
1. **Trapezoid:**
- Coordinates: Height = 80 mm, Bases = 45 mm and 25 mm, Other side = 65 mm
2. **Rectangle:**
- Coordinates: W = 55 mm, H = 80 mm
3. **Semi-circle:**
- Diameter = 65 mm (radius = 32.5 mm)
Additionally, the trapezoid and rectangle are placed on a baseline where these dimensions become critical for calculating their centroid location.
**Centroid Calculation:**
The centroid \(\bar{x}\) and \(\bar{y}\) can be computed using the formula for composite areas, taking the moments of individual areas about the baseline (x, y reference axes):
1. Break down the composite area into distinct, simpler shapes (trapezoid, rectangle, and semi-circle).
2. Calculate the area (A) and centroid (\(x\), \(y\)) of each simpler shape.
3. Use the formula:
\[ \bar{x} = \frac{\Sigma(A_i \cdot x_i)}{\Sigma A_i} \]
\[ \bar{y} = \frac{\Sigma(A_i \cdot y_i)}{\Sigma A_i} \]
Where:
- \(A_i\) = Area of the ith shape
- \(x
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