# HELP! Related Rates Question: Light/Shadow

#### jen333

Hey, I have this one practise calculus question that I just can't seem to get. Any help would be greatly appreciated:

A light is on the ground 40ft from a building. A man 6ft tall walks from the light towards the building at 6ft/s. How rapidly is his shadow on the building becoming shorter when he is 20ft from the building?
(btw, the answer should be -3.6ft/s)

I've already drawn a diagram, but I'm not sure if triangle ratios would do any good.

(oops, wrong forum AGAIN! I'd delete this is i could. Refer to calculus and beyond. Sorry)

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#### HallsofIvy

jen333 said:
Hey, I have this one practise calculus question that I just can't seem to get. Any help would be greatly appreciated:

A light is on the ground 40ft from a building. A man 6ft tall walks from the light towards the building at 6ft/s. How rapidly is his shadow on the building becoming shorter when he is 20ft from the building?
(btw, the answer should be -3.6ft/s)

I've already drawn a diagram, but I'm not sure if triangle ratios would do any good.

(oops, wrong forum AGAIN! I'd delete this is i could. Refer to calculus and beyond. Sorry)
I'll transfer it for you.

I just wrote out a complete solution thinking the light was on the building and his shadow on the ground!! Anyway the ratios are what you want but be careful about exactly what they are. Let x be the length of the man's shadow, on the building, and y be his distance from the light.
You have two similar right triangles:
1) The triangle formed by the man, line from the man to the light, and the hypotenuse. The ratio of second leg to first is y/6.
2) The triangle formed by the shadow of the man on the building, the line from the light to the base of the building, and its hypotenuse. The corresponding ratio is 40/x.
Since those triangles are similar, y/6= 40/x or xy= 240. Differentiating both sides, x'y+ xy'= 0. You are told that y'= 8 and you want to find x' when y= 20. Of course, then 20x= 240 so x= 12. 20x'+ 12(6)= 0. Solve for x'.

Halls, I thought you werent supposed to give complete solutions...........I had written on this already giving her a hint... Also that is making it way too complicated. just write y = 240/x, and differentiate from there and plug in the values.

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#### jen333

Thanx

Oh, hahaha. don't worry. I solved it before I saw the written out solution. Thanks so much for your help :D

-jen

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