Finding Arc Length in Optimization Problem

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Homework Statement


Joe is traveling from point A across a circular lake to a cabin on the other side at point B. The straight line distance from A to B is 3 miles and is the diameter of the lake. He travels in a canoe on a straight line from A to C. She then takes the circular trail from C to B. She can travel along the circular trail at 4 mph and paddle in the canoe at 2mph.

There is also a picture provided. It is basically a circle with an scalene triangle inscribed in it. on leg is from A to B (diameter). Another leg is from A to C. And finally the last leg is C to B. Only angle provided is theta which is the angle from C to B.

a) Determine an expression in terms of theta for the length of arc CB.

b) Determine an expression in terms of theta for the length of segment AC.

c) Determine an expression in terms of theta for the total time travelling along AC and CB.




The Attempt at a Solution



I basically started by splitting the triangle into two triangles. I did this by making a line from the midpoint of the diameter to the point C. i know that arc length is basically circumference multiplied by theta/360 but im having a difficult time understanding how to make an expression for that.
 

Answers and Replies

  • #2
Mentallic
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An expression for the arc length from A to B (denoted sAB) would be

[tex]s_{AB} = \frac{C\cdot \theta}{360}[/tex]

Where [itex]\theta = 180[/itex] and [itex]C = 2\pi r[/itex] and [itex]r=1.5mi[/itex]

So

[tex]s_{AB} \approx 4.71mi[/tex]

Now just find the expression for the arc length BC (denoted sBC) and leave it in terms of [itex]\theta[/itex]
 
  • #3
Mentallic
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Fullmetalx said:
thanks for replying. it seems like not on of my classmates knows how to do this problem. im not quite understanding how you are getting 4.71 miles.

here is a pic of what it looks like
http://img33.imageshack.us/img33/446/photolwo.jpg [Broken]

Fullmetalx, that link is broken, and the 4.71 miles is the arc length from A to B which is half way around the circle. It was just an example to help get you started with finding the arc length from B to C, which you need to denote in terms of [itex]\theta[/itex].

[tex]s_{AB} = \frac{1.5\cdot 2\pi\theta}{360}[/tex]

where in this case, [itex]\theta = 180[/itex] which I then simplified into [itex]s_{AB}=1.5\pi\approx 4.71[/itex] so sBC is the same formula, but for an variable value of [itex]\theta[/itex].

Now, do you know how to find the length of the segment AC? You must have a formula given to you in class, but it's also not too difficult to derive for yourself.
 
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