1. **Problem statement:** A man walks along a path from a house at 2 m/s. The light is 7 m from the house and 5 m from the path. We want to find the rate at which his shadow moves along the house wall when he is 3 m from the house. 2. **Setup and variables:** Let $x$ be the distance of the man from the house along the path, and $s$ be the length of his shadow on the house wall. 3. **Known values:** - Distance from light to house: 7 m - Distance from light to path: 5 m - Man's speed: $\frac{dx}{dt} = 2$ m/s - Position of man: $x = 3$ m 4. **Using similar triangles:** The light, the man, and the shadow form similar triangles. The ratio of the shadow length $s$ to the distance from the light to the path (5 m) equals the ratio of the man's distance from the light to the distance from the light to the house (7 m + $x$): $$\frac{s}{5} = \frac{7 + x}{7}$$ 5. **Solve for $s$:** $$s = 5 \cdot \frac{7 + x}{7} = \frac{5}{7}(7 + x) = 5 + \frac{5}{7}x$$ 6. **Differentiate with respect to time $t$:** $$\frac{ds}{dt} = \frac{5}{7} \frac{dx}{dt}$$ 7. **Substitute $\frac{dx}{dt} = 2$ m/s:** $$\frac{ds}{dt} = \frac{5}{7} \times 2 = \frac{10}{7} \approx 1.43 \text{ m/s}$$ 8. **Interpretation:** The shadow moves along the wall at approximately 1.43 meters per second when the man is 3 meters from the house. **Final answer:** $$\boxed{\frac{ds}{dt} = \frac{10}{7} \text{ m/s} \approx 1.43 \text{ m/s}}$$