Hello, jmailloux!

A man 2 m tall walks away from a lamppost whose light is 5 m above the ground.

If he walks at a speed of 1.5 m/s, at what rate is his shadow increasing

when he is 10 m from the lamppost.

The answer is supposed to be 1 m/s, but I keep on getting 2.5 m/s,

and I have no clue what I'm doing wrong. . I know!

You made a very common error.

I know ... I still do it from time to time. Code:

*
| *
| *
| *
5| | *
| 2| *
| | *
* - - - - - + - - - - - - - *
: x : s :

I bet you let $\displaystyle s$ equal the whole bottom of the diagram.

That would give us the rate at which the tip of his shadow is moving.

. . But they want the rate at which the *length of his shadow* is changing.

From the similar right triangles, we have: .$\displaystyle \frac{s}{2}\:=\:\frac{x+s}{5}$

Then: .$\displaystyle 5s\:=\:2x + 2s\quad\Rightarrow\quad 3s \,= \,2x\quad\Rightarrow\quad s \,= \,\frac{2}{3}x$

Differentiate with respect to time: .$\displaystyle \frac{ds}{dt}\:=\:\frac{2}{3}\cdot\frac{dx}{dt}$

The man's speed is 1.5 m/sec: .$\displaystyle \frac{dx}{dt} = 1.5$

Therefore: .$\displaystyle \frac{ds}{dt}\:=\:\frac{2}{3}(1.5) \:=\:1\text{ m/s}$