1. **State the problem:** We need to find the length of the elevator path that goes up the incline of a lateral edge of a pyramid with a rectangular base of dimensions 80 m by 60 m and a height of 150 m. 2. **Understand the pyramid structure:** The pyramid has a rectangular base with length 80 m and width 60 m, and a vertical height of 150 m. 3. **Find the coordinates of the apex and base corners:** - Let the base be on the xy-plane with corners at (0,0,0), (80,0,0), (80,60,0), and (0,60,0). - The apex is at the center of the base in xy-plane and height 150 m, so its coordinates are at the midpoint of the base: $$\left(\frac{80}{2}, \frac{60}{2}, 150\right) = (40, 30, 150)$$. 4. **Find the length of the lateral edge:** The elevator path goes along a lateral edge, which connects the apex to one of the base corners. 5. **Calculate the distance from apex to a base corner:** Use the distance formula in 3D: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$ Choose the corner at (0,0,0): $$d = \sqrt{(40 - 0)^2 + (30 - 0)^2 + (150 - 0)^2} = \sqrt{40^2 + 30^2 + 150^2}$$ 6. **Calculate the squares:** $$40^2 = 1600$$ $$30^2 = 900$$ $$150^2 = 22500$$ 7. **Sum the squares:** $$1600 + 900 + 22500 = 25000$$ 8. **Find the square root:** $$d = \sqrt{25000} = 50\sqrt{10} \approx 158.11$$ meters **Final answer:** The length of the elevator path along the lateral edge is approximately **158.11 meters**.