Question: Determine the centroid of the composite shape.
Shape description for computation:
Composite polygon with x-y axes at the bottom-left corner; the shape can be divided into 4 regions: a left triangle (region 1) with base $2$ m and height $6.5$ m, a central rectangle/trapezoid-like block (region 2) from $x=2$ m to $x=4$ m rising to the top, a right triangle (region 3) sloping down from the top-right of region 2 to the point at $x=8$ m, $y=1$ m, and a bottom-right rectangle (region 4) from $x=4$ m to $x=9$ m with height $1$ m.
position_hint: centerUser: answer the centroid trough table form
1. **State the problem:** Find the centroid $(\bar{x}, \bar{y})$ of the composite shape made of 4 regions.
2. **Identify each region and its properties:**
- Region 1: Left triangle with base $b_1=2$ m and height $h_1=6.5$ m.
- Region 2: Central trapezoid-like block from $x=2$ m to $x=4$ m, height $h_2=6.5$ m (same top height as region 1).
- Region 3: Right triangle sloping down from $(4,6.5)$ to $(8,1)$.
- Region 4: Bottom-right rectangle from $x=4$ m to $x=9$ m with height $h_4=1$ m.
3. **Calculate area and centroid of each region:**
- Region 1 (triangle):
$$A_1 = \frac{1}{2} b_1 h_1 = \frac{1}{2} \times 2 \times 6.5 = 6.5$$
Centroid coordinates:
$$x_1 = \frac{b_1}{3} = \frac{2}{3} \approx 0.667$$
$$y_1 = \frac{h_1}{3} = \frac{6.5}{3} \approx 2.167$$
- Region 2 (rectangle):
Width $w_2 = 4 - 2 = 2$ m, height $h_2 = 6.5$ m
$$A_2 = w_2 \times h_2 = 2 \times 6.5 = 13$$
Centroid:
$$x_2 = 2 + \frac{w_2}{2} = 2 + 1 = 3$$
$$y_2 = \frac{h_2}{2} = \frac{6.5}{2} = 3.25$$
- Region 3 (triangle):
Base $b_3 = 8 - 4 = 4$ m, height difference $h_3 = 6.5 - 1 = 5.5$ m
$$A_3 = \frac{1}{2} b_3 h_3 = \frac{1}{2} \times 4 \times 5.5 = 11$$
Centroid coordinates (for a right triangle with right angle at $(4,6.5)$ and sloping down to $(8,1)$):
$$x_3 = 4 + \frac{b_3}{3} = 4 + \frac{4}{3} = 5.333$$
$$y_3 = 1 + \frac{h_3}{3} = 1 + \frac{5.5}{3} = 2.833$$
- Region 4 (rectangle):
Width $w_4 = 9 - 4 = 5$ m, height $h_4 = 1$ m
$$A_4 = w_4 \times h_4 = 5 \times 1 = 5$$
Centroid:
$$x_4 = 4 + \frac{w_4}{2} = 4 + 2.5 = 6.5$$
$$y_4 = \frac{h_4}{2} = 0.5$$
4. **Calculate total area:**
$$A = A_1 + A_2 + A_3 + A_4 = 6.5 + 13 + 11 + 5 = 35.5$$
5. **Calculate composite centroid coordinates:**
$$\bar{x} = \frac{\sum A_i x_i}{A} = \frac{6.5 \times 0.667 + 13 \times 3 + 11 \times 5.333 + 5 \times 6.5}{35.5}$$
Calculate numerator:
$$6.5 \times 0.667 = 4.3355$$
$$13 \times 3 = 39$$
$$11 \times 5.333 = 58.663$$
$$5 \times 6.5 = 32.5$$
Sum:
$$4.3355 + 39 + 58.663 + 32.5 = 134.4985$$
$$\bar{x} = \frac{134.4985}{35.5} \approx 3.79$$
Similarly for $\bar{y}$:
$$\bar{y} = \frac{6.5 \times 2.167 + 13 \times 3.25 + 11 \times 2.833 + 5 \times 0.5}{35.5}$$
Calculate numerator:
$$6.5 \times 2.167 = 14.0855$$
$$13 \times 3.25 = 42.25$$
$$11 \times 2.833 = 31.163$$
$$5 \times 0.5 = 2.5$$
Sum:
$$14.0855 + 42.25 + 31.163 + 2.5 = 90.0$$
$$\bar{y} = \frac{90.0}{35.5} \approx 2.54$$
6. **Summary table:**
| Region | Shape | Area $A_i$ (m$^2$) | $x_i$ (m) | $y_i$ (m) |
|--------|-----------|--------------------|-----------|-----------|
| 1 | Triangle | 6.5 | 0.667 | 2.167 |
| 2 | Rectangle | 13 | 3 | 3.25 |
| 3 | Triangle | 11 | 5.333 | 2.833 |
| 4 | Rectangle | 5 | 6.5 | 0.5 |
7. **Final centroid coordinates:**
$$\boxed{(\bar{x}, \bar{y}) = (3.79, 2.54)}$$
This is the centroid of the composite shape measured from the bottom-left corner (origin).