A calculus question involving derivatives, and rates of change when a ratio exists.?

Question

Here is the question: "A man 6 feet tall walks at a rate of 5 feet per second away from a light that is 15 feet above the ground. When he is 10 feet from the base of the light, a) at what rate is the tip of the shadiw moving? and b) at what rate is the length of his shadow changing?

It is a triangle with a ratio between the light and the person (height). Then there are distances that exist between the person and the light, and the person and the tip of his shadow. The height of the light, again, is 15. The value of him going away from the light is 10. He is 6 (feet tall).

This will involve (dy/dt) somehow, or some equivalent. Best answer must include details to arrive at the correct answer. Remember, this is a two part-question. A hint was given, saying that doing part b) first will make part a) easier because you will have already solved for something. Be sure to label the parts, so I know where you are in explanation.

Thanks!

Answer 1

(b) You must understand that there are two similar triangles here in order to set up the first relation. The first triangle is the man's shadow(base) and his body(height) (THE RIGHT HAND SIDE OF THE EQUATION). The second triangle is the lamp post(height) and the space between the lamp post plus the man's shaddow(base).

So by similar triangles, we have ...

[x(t) + y(t)] / 15ft = x(t) / 6ft

1/15 (dx/dt + dy/dt) = 1/6 dx/dt

we know that dy/dt = 5ft/s

1/15 (dx/dt + 5) = 1/6 dx/dt

1/10 (dx/dt) = 1/3

dx/dt = 3.33 ft/sec

(a) Part a should definitely be done second because it goes as follows ...

We now know the speed of shadow growth, so how fast does this tip move?

Speed of shadow growth = Speed of tip - Speed of Man

3.3 ft/s = Speed of tip - 5 ft/s

Speed of tip = 8.3 ft/s

Hope this helps. If you need any clarification, just message me.