Composite Area Centroid 62Bd70

1. **State the problem:** We have a composite shape consisting of a large rectangle with two notches: a rectangular notch on the upper-left and a semicircular notch on the upper-right. We need to find the total area and the coordinates of the centroid $(\bar{x}, \bar{y})$. 2. **Identify dimensions and shapes:** - Large rectangle: width $= 8$ m (from $-4$ to $4$), height $= 4$ m. - Rectangular notch: width $= 2$ m, height $= 2$ m, located at upper-left corner. - Semicircular notch: diameter $= 2$ m, radius $r = 1$ m, located at upper-right corner. 3. **Calculate areas:** - Area of large rectangle: $$A_1 = 8 \times 4 = 32 \text{ m}^2$$ - Area of rectangular notch: $$A_2 = 2 \times 2 = 4 \text{ m}^2$$ - Area of semicircular notch: $$A_3 = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (1)^2 = \frac{\pi}{2} \approx 1.5708 \text{ m}^2$$ 4. **Calculate total area:** $$A = A_1 - A_2 - A_3 = 32 - 4 - \frac{\pi}{2} = 28 - \frac{\pi}{2} \approx 26.4292 \text{ m}^2$$ 5. **Find centroids of each shape:** - Large rectangle centroid $(x_1, y_1)$ is at center: $$x_1 = 0, \quad y_1 = \frac{4}{2} = 2$$ - Rectangular notch centroid $(x_2, y_2)$ is at its center: - Horizontally, notch extends from $x = -4$ to $x = -2$, so centroid $x_2 = \frac{-4 + (-2)}{2} = -3$ - Vertically, notch extends from $y = 2$ to $y = 4$, so centroid $y_2 = \frac{2 + 4}{2} = 3$ - Semicircular notch centroid $(x_3, y_3)$: - Horizontally, notch extends from $x = 2$ to $x = 4$, so center of full circle is at $x = 3$ - Vertically, notch is at top edge $y = 4$ - Centroid of a semicircle from flat edge is at a distance $\frac{4r}{3\pi}$ from the flat edge inside the shape. - Since notch is cut out from top, centroid of notch area is below the top edge by: $$d = \frac{4r}{3\pi} = \frac{4 \times 1}{3\pi} = \frac{4}{3\pi} \approx 0.4244$$ - So $y_3 = 4 - d = 4 - 0.4244 = 3.5756$ 6. **Calculate moments for centroid:** - Total area $A = A_1 - A_2 - A_3$ - $\bar{x} = \frac{A_1 x_1 - A_2 x_2 - A_3 x_3}{A}$ - $\bar{y} = \frac{A_1 y_1 - A_2 y_2 - A_3 y_3}{A}$ Calculate numerator for $\bar{x}$: $$32 \times 0 - 4 \times (-3) - \frac{\pi}{2} \times 3 = 0 + 12 - \frac{3\pi}{2} = 12 - 4.7124 = 7.2876$$ Calculate numerator for $\bar{y}$: $$32 \times 2 - 4 \times 3 - \frac{\pi}{2} \times 3.5756 = 64 - 12 - 2.8125 = 64 - 12 - 2.8125 = 49.1875$$ 7. **Calculate centroid coordinates:** $$\bar{x} = \frac{7.2876}{28 - \frac{\pi}{2}} = \frac{7.2876}{26.4292} \approx 0.276$$ $$\bar{y} = \frac{49.1875}{26.4292} \approx 1.861$$ **Final answers:** - Area: $$\boxed{26.43 \text{ m}^2}$$ - Centroid $x$-coordinate: $$\boxed{0.28 \text{ m}}$$ - Centroid $y$-coordinate: $$\boxed{1.86 \text{ m}}$$