Man walking away from a lamp post: A lamp post is 5 m high. A 2 m tall man walks directly away from the post at 6 m/min. At what rate is the length of his shadow increasing?

Introduction / Context:
Similar triangles relate the lamp, the man, and the tip of the shadow. Differentiating this relation with respect to time gives the rate of change of the shadow length.

Given Data / Assumptions:

• Lamp height = 5 m; man height = 2 m.
• Man walks away from the lamp at dx/dt = 6 m/min.
• Let shadow length be s (from man to tip).

Concept / Approach:
The geometry yields a linear relation between s and the distance x of the man from the lamp. Then compute ds/dt via differentiation.

Step-by-Step Solution:

Verification / Alternative check:
Pick x = 30 m → s = 20 m. After 1 min, x increases by 6 m → s increases by 4 m, matching the derivative.

Why Other Options Are Wrong:
8 and 9 m/min contradict the linear 2/3 factor; 14 m/min is non-physical for these heights and speed.

Common Pitfalls:
Using 5/x = 2/s (omitting x + s); differentiating incorrectly; forgetting that s is measured from the man to the tip, not from the lamp.

Final Answer:
4 m/min