Related Rates involving Cosine Law

[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!

 

[STEMucator]

Edit: Nvm

 

[LCKurtz]

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

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.

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.

Thanks!

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 $T$.