Final answer:
Using similar triangles and calculus, the rate at which the shadow lengthens is found to be 0.20 m/s while the tip of the shadow moves at a rate of 0.50 m/s as a man walks away from a streetlight.
Step-by-step explanation:
The question is concerned with the rate at which a man's shadow is moving and lengthening as he walks away from a streetlight. To resolve the problem, we use the concepts of similar triangles and rates of change, which are topics in geometry and calculus respectively. Let's denote the height of the streetlight as L, the height of the man as h, his distance from the streetlight as x, and the length of the shadow as y. Given that L = 4 m, h = 1.6 m, and the man's speed is 0.3 m/s, we can set up a proportion because the triangles formed are similar:
L / (x + y) = h / y. Differentiating both sides with respect to time, we get: L * (dx/dt + dy/dt) = h * dy/dt. Substituting the values and solving for dy/dt (rate at which the shadow lengthens) and dx/dt + dy/dt (rate at which the tip of the shadow moves) gives us 0.20 m/s for dy/dt and 0.50 m/s for dx/dt + dy/dt.