Related Rates involving Cosine Law

  • #1
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[itex]\theta[/itex]
A = [itex]\pi[/itex]r^2
C = [itex]\pi[/itex]d = 2[itex]\pi[/itex]r
A of Sector= 1/2([itex]\theta[/itex]-Sin[itex]\theta[/itex])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[itex]\theta[/itex]
So: dL/dt = dr/dt x d[itex]\theta[/itex]/dt
I assumed the walking speed on the surface was dL/dt. Therefore, dL/dt = 4.
I assumed d[itex]\theta[/itex]/dt = 4 m/s x 2[itex]\pi[/itex] 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 [itex]\theta[/itex] 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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  • #2
STEMucator
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Edit: Nvm
 
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  • #3
LCKurtz
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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[itex]\theta[/itex]
A = [itex]\pi[/itex]r^2
C = [itex]\pi[/itex]d = 2[itex]\pi[/itex]r
A of Sector= 1/2([itex]\theta[/itex]-Sin[itex]\theta[/itex])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[itex]\theta[/itex]
So: dL/dt = dr/dt x d[itex]\theta[/itex]/dt
I assumed the walking speed on the surface was dL/dt. Therefore, dL/dt = 4.
I assumed d[itex]\theta[/itex]/dt = 4 m/s x 2[itex]\pi[/itex] 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 [itex]\theta[/itex] 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##.
 

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