(a)The arc lamp to the shadow, and the man's head to the shadow form two similar triangles.Originally Posted byxiaoz

The ratio of height to base for the man's triangle is

The ratio of height to base for the arc-lamp triangle is

Because the triangles are similar, the two ratios are equal...

Which means the height of the arc-lamp triangle is 9m.

(b) This is an interesting question. Imagine the man on a co-ordinate plane with the arc-lamp in the center. He is on point (10,0) and walks to point (10,20).

We need to first find the distance between him and the lamp. The distance formula is...

note:

Just to make it easy to write, we'll say the distance between the man and the lamp is

Now we know the heights of both triangles, we know they're similar, so we remake our ratios...

and solve from there.