Question

A 5.5​-foot-tall woman walks at 4 ​ft/s toward a street light that is 27.5 ft above the ground. what is the rate of change of the length of her shadow when she is 16 ft from the street​ light? at what rate is the tip of her shadow​ moving?

Answer 1

Rate at which the shadow is moving = -5 ft/s.

Explanation:

I can do the second part for you:

If the distance of the woman from the wall is Xp , the length of the shadow is Xs and the distance from the tip of the shadow to the wall is X we have the relation:

X = Xp + Xs.

We need to find X' (the rate that the tip of the shadow is moving). at Xp = 16 and X'p = -4 ft/s.

We need a relation between X and Xp so we have to eliminate Xs.

By similar triangles 5.5 / 27.5 = Xs / x

1/5 = Xs/x

Xs = x /5 so substituting in the above relation:

X = Xp + X/5

4X/5 = Xp

X = 5Xp / 4

Taking derivatives:

X' = 5X'p / 4

Now X'p is given as - 4 so

X' = -20/4 = -5 ft/s.