The question is as follows:
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 his shadow moving?
(b) at what rate is the length of his shadow changing?
Answer key with visual http://c782279.r79.cf2.rackcdn.com/se02f01033.png
My question isn't about the process of finding the answers, but more about why the answer key placed the variables where they did. Currently, the distance from the man to the light post is assigned the variable x, the distance from the tip of the shadow to the light post is assigned y, and the distance from the tip to the man is (y-x).
I tried to complete the problem with the distance from the man to the light post designated as x, the distance from the tip of the shadow to the man as y, and the distance from the tip of the shadow to the light post as x + y. After doing the calculations for part (a) several times, I found that I was unable to obtain the correct answer.
I asked an online tutor why the variables had to be placed in the locations on the answer key, and he told me that it was because when differentiating to find the answer in part (a), you could not differentiate (x+y) because it had to be a specific variable.
While this answer does make sense to me, I can remember my teacher from last year giving a more logical explanation. If I remember correctly, he said that x and y represented variables that could independently changed, while y-x represented a value that couldn't be changed on its own.
I thought about this explanation for a long time, and couldn't figure out why the distance from the tip of the shadow to the man was dependent while the distance from the tip of the shadow to the light post was independent. If my explanation is being incorrectly remembered, could you please correct it, and if there is some other fundamental reason that I am overlooking, could you please explain that?
Best Answer
There is no reason why you can't set the variables as you did, and why you should not get the right answer if you proceed properly !
By similar triangles, $\dfrac{y}{6} = \dfrac{x}{15-6}$ which yields $y = \dfrac{2x}{3}$
so, (b) $\dfrac{dy}{dt} = \dfrac23 \cdot \dfrac{dx}{dt} =\dfrac{10}{3}$ ft/s
and (a), $\dfrac{dx}{dt} + \dfrac{dy}{dt} = \dfrac{25}{3}$ ft/s
In fact, this is my preferred method, which gets to the answers more simply. [ Compare the initial algebraical manipulation needed for the two methods ]