Optimization Problem Please help

[dhaliwalia]

A man wants to get the other side of a circular lake of radius 300 ft. He
can walk all the way around, or he could swim all the way across, or he could
pick an angle  to swim to another point on the shore, then walk the rest of the
way. If he walks at 8 ft/sec, and swims at 4 ft/sec. Find the angles which (a)
minimize travel time, and (b) maximize travel time. Find the time of travel in
each case.

 

[Deleted member 4993]

A man wants to get the other side of a circular lake of radius 300 ft. He can walk all the way around, or he could swim all the way across, or he could pick an angle to swim to another point on the shore, then walk the rest of the way. If he walks at 8 ft/sec, and swims at 4 ft/sec. Find the angles which (a) minimize travel time, and (b) maximize travel time. Find the time of travel in each case.

Hint:

First draw a sketch of the situation.

Please share your work with us, indicating exactly where you are stuck - so that we may know where to begin to help you.

 

[pappus]

A man wants to get the other side of a circular lake of radius 300 ft. He
can walk all the way around, or he could swim all the way across, or he could
pick an angle to swim to another point on the shore, then walk the rest of the
way. If he walks at 8 ft/sec, and swims at 4 ft/sec. Find the angles which (a)
minimize travel time, and (b) maximize travel time. Find the time of travel in
each case.

Second hint:

There are 3 ways to consider!

 

[dhaliwalia]

I got farther than that on my own. Starting is easy, I need that total distance and time function.

 

[galactus]

Create an expression for the distance swam and one for the distance walked.

The distance walked around the edge could be \(\displaystyle r\cdot 2\alpha\)

The distance swam can be found by the law of cosines where the angle would be

\(\displaystyle \pi-2\alpha\). So, \(\displaystyle cos(\pi-2\alpha)=-cos2\alpha\)

 

[pappus]

I got farther than that on my own. <-- interesting! And why didn't you tell us how far you got? It would have saved us a lot of time ...
Starting is easy, I need that total distance and time function.

You certainely have noticed:

............. that triangle SPF is a right triangle;

............. that \(\displaystyle s = 2 r \cdot \cos(\alpha) \):

............. that \(\displaystyle w = 2 \alpha \cdot r \);

............. that you have to use the definition of speed to find the elapsed times for the distances s and w.

Obviously the time function is:

\(\displaystyle \displaystyle{ \)\(\displaystyle t(\alpha) = \frac{s}{4} + \frac{w}{8} } \)

t(alpha) = s/4 + w/8

... and now I hope that this post contains something new for you.


EDIT: Not sure why LaTeX isn't working properly now. I've repeated the equation of the time function in plain text.