Related Rates with Shadows and Similar Triangles

Picture the lamppost as the tall side of one right triangle and the person as the tall side of a smaller triangle inside it. Both triangles share the same light ray as their hypotenuse, so they're similar: the ratio of height to base is the same for each. If the lamppost has height H, the person has height h, the person stands a distance x from the pole, and the shadow has length s, then the big triangle has height H and base x+s, while the small triangle has height h and base s. Similar triangles give H/(x+s) = h/s, which simplifies to a clean relationship between x and s with no square roots involved.

Once you have that equation, solve it for s in terms of x: s = hx/(H-h). Since h and H never change, differentiating is just applying a constant multiple: ds/dt = (h/(H-h)) dx/dt. Plug in the person's walking speed for dx/dt and you have how fast the shadow itself is stretching. The shadow's tip is a different object entirely — its distance from the pole is x+s, not s — so its speed is dx/dt + ds/dt, always faster than the shadow's growth alone.

This sits next to the ladder and wall problem as the two classic related rates setups: similar triangles when the geometry is two nested triangles sharing an angle, and the Pythagorean theorem when the geometry is a right triangle with a fixed hypotenuse or fixed leg. Recognizing which picture you're in — triangles sliding inside each other, or a fixed-length segment sweeping — tells you which equation to differentiate.