1. **Problem statement:** Dr. Acula jogs around a cemetery forming a polygon with the following path: 3.5 km east, 5 km northeast, 5 km northwest, 5 km southwest, and 3.5 km south, returning to the start. We need to find the area of this polygon. 2. **Approach:** We will find the coordinates of each vertex assuming the start point is at the origin $(0,0)$, then use the shoelace formula to find the polygon's area. 3. **Coordinates calculation:** - Start at $A = (0,0)$. - Move 3.5 km east: $B = (3.5,0)$. - Move 5 km northeast (45°): $C = (3.5 + 5\cos 45^\circ, 0 + 5\sin 45^\circ) = (3.5 + 5\times \frac{\sqrt{2}}{2}, 5\times \frac{\sqrt{2}}{2}) = (3.5 + 3.5355, 3.5355) = (7.0355, 3.5355)$. - Move 5 km northwest (135°): $D = (7.0355 + 5\cos 135^\circ, 3.5355 + 5\sin 135^\circ) = (7.0355 - 3.5355, 3.5355 + 3.5355) = (3.5, 7.071)$. - Move 5 km southwest (225°): $E = (3.5 + 5\cos 225^\circ, 7.071 + 5\sin 225^\circ) = (3.5 - 3.5355, 7.071 - 3.5355) = (-0.0355, 3.5355)$. - Move 3.5 km south: back to $A = (0,0)$, which checks out since $(-0.0355, 3.5355 - 3.5) = (-0.0355, 0.0355)$ is close to zero due to rounding. 4. **Shoelace formula:** For polygon vertices $(x_1,y_1), (x_2,y_2), ..., (x_n,y_n)$, area = $$\frac{1}{2} \left| \sum_{i=1}^{n-1} (x_i y_{i+1} - x_{i+1} y_i) + (x_n y_1 - x_1 y_n) \right|$$ 5. **Apply shoelace formula:** - Vertices: $A(0,0), B(3.5,0), C(7.0355,3.5355), D(3.5,7.071), E(-0.0355,3.5355)$ Calculate: $$\sum x_i y_{i+1} = 0\times0 + 3.5\times3.5355 + 7.0355\times7.071 + 3.5\times3.5355 + (-0.0355)\times0 = 0 + 12.374 + 49.74 + 12.374 + 0 = 74.488$$ $$\sum y_i x_{i+1} = 0\times3.5 + 0\times7.0355 + 3.5355\times3.5 + 7.071\times(-0.0355) + 3.5355\times0 = 0 + 0 + 12.374 - 0.251 + 0 = 12.123$$ Area = $$\frac{1}{2} |74.488 - 12.123| = \frac{1}{2} \times 62.365 = 31.1825$$ 6. **Final answer:** The area is approximately $31$ km$^2$. **Answer:** 31 km$^2$