Determine centroid X and Y of the shaded area shown.
The Correct Answer and Explanation is :
To determine the centroid (center of mass) of the shaded area shown in the provided image, we can apply the principles of composite areas. The dimensions given are 2 ft × 2 ft, 3 ft × 1 ft, and 2 ft × 3 ft. citeturn0search5
Step 1: Break the Shape into Simpler Components
The shaded area can be decomposed into three rectangles:
- Rectangle A: Dimensions 2 ft × 2 ft
- Rectangle B: Dimensions 3 ft × 1 ft
- Rectangle C: Dimensions 2 ft × 3 ft
Step 2: Calculate the Area and Centroid of Each Rectangle
- Rectangle A:
- Area (A₁): 2 ft × 2 ft = 4 ft²
- Centroid coordinates (x₁, y₁): (1 ft, 1 ft)
- Located at half the length and half the width.
- Rectangle B:
- Area (A₂): 3 ft × 1 ft = 3 ft²
- Centroid coordinates (x₂, y₂): (4.5 ft, 0.5 ft)
- Placed 1.5 ft from the left edge and 0.5 ft from the bottom edge.
- Rectangle C:
- Area (A₃): 2 ft × 3 ft = 6 ft²
- Centroid coordinates (x₃, y₃): (1 ft, 4 ft)
- Situated 1 ft from the left edge and 4 ft from the bottom edge.
Step 3: Compute the Weighted Average of Centroids
The overall centroid (X, Y) of the composite shape is found by calculating the weighted average of the centroids of the individual rectangles, weighted by their respective areas:
- Total Area (A):
- A = A₁ + A₂ + A₃ = 4 ft² + 3 ft² + 6 ft² = 13 ft²
- X-coordinate of Centroid (X):
- X = (A₁ × x₁ + A₂ × x₂ + A₃ × x₃) / A
- X = (4 ft² × 1 ft + 3 ft² × 4.5 ft + 6 ft² × 1 ft) / 13 ft²
- X = (4 ft² + 13.5 ft² + 6 ft²) / 13 ft²
- X = 23.5 ft² / 13 ft²
- X ≈ 1.81 ft
- Y-coordinate of Centroid (Y):
- Y = (A₁ × y₁ + A₂ × y₂ + A₃ × y₃) / A
- Y = (4 ft² × 1 ft + 3 ft² × 0.5 ft + 6 ft² × 4 ft) / 13 ft²
- Y = (4 ft² + 1.5 ft² + 24 ft²) / 13 ft²
- Y = 29.5 ft² / 13 ft²
- Y ≈ 2.27 ft
Conclusion
The centroid of the shaded area is located approximately at coordinates (X, Y) = (1.81 ft, 2.27 ft) relative to the bottom-left corner of the shape.