Related Rates involving Cosine Law

1. Jun 15, 2014

Cardinality

For problem: See Attachment

I've never done a problem of this sort and it's proving to be much more difficult compared to the other problems I have had assigned to me.
I'm not entirely sure which formulas to use but I've been playing with the following:
Length of Arc = r$\theta$
A = $\pi$r^2
C = $\pi$d = 2$\pi$r
A of Sector= 1/2($\theta$-Sin$\theta$)r^2
And the Cosine Law

Being honest, I'm a little confused by the question. I'm not entirely sure if I labelled my variables correctly. But I made a diagram and this is what I did so far:

- L = r$\theta$
So: dL/dt = dr/dt x d$\theta$/dt
I assumed the walking speed on the surface was dL/dt. Therefore, dL/dt = 4.
I assumed d$\theta$/dt = 4 m/s x 2$\pi$ rad/m = 8.
Therefore, I was able to calculate that dr/dt = 1/2.

I differentiated the cosine formula just to give myself a general idea of what I had to work with, but with two variable side lengths and an unknown $\theta$ I was at a loss.

Note: After quite a bit of searching, this was the closest value I found to help me out.

But in this example, a value for theta is given and two side lengths are known.

I think my main problem may be understanding the question. If someone could clarify it and give me some pointers that would be great!

Thanks!

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Last edited by a moderator: Sep 25, 2014
2. Jun 15, 2014

Zondrina

Edit: Nvm

Last edited: Jun 15, 2014
3. Jun 15, 2014

LCKurtz

It is pretty pointless just to write down formulas willy-nilly. You need to draw a picture and figure out a formula for what you are trying to maximize or minimize. And this problem has nothing to do with areas.

Without seeing your diagram, how are we supposed to follow that? And given that the radius of the circle is 20, clearly finding dr/dt = 1/2 is nonsense. Here's what I suggest you do:

1. Draw a circle at the origin with A at (20,0) and B at (-20,0). Label a point on the upper circumference as C with polar angle $\theta$. Label the arc from A to C as $s$ and the straight line from C to B as $l$. So the runner/swimmer is going to run around the circle from A to the unknown point C along $s$, then swim directly from C to B along $l$.

2. Write an equation for the time $T$ it takes for him to run/swim that path. Only then are you ready to differentiate with respect to time $t$ and minimize $T$.