Optimization

[legacyofpiracy]

Hello, I am at a loss as to how to begin on this problem using optimization. There are no examples in the book and we did not cover this in class. Any help would be greatly appreciated.

the problem:

Jane is 2 miles offshore in a boat and wishes to reach a costal village 6 miles down a straight shoreline from the point nearest the boat. She can row 2 mph and can walk 5 mph. Where should she land her boat to reach the village in the least amount of time?

Any suggestions?

 

[soroban]

Hello, legacy!

Did you make a sketch?

Jane is 2 miles offshore in a boat and wishes to reach a costal village 6 miles down
a straight shoreline from the point nearest the boat.
She can row 2 mph and can walk 5 mph.
Where should she land her boat to reach the village in the least amount of time?

. . . J
. . . *
. . . | \                                Jane is at J.
. . . | . \    ______                The village is at V.
. . 2 | . . \ √x² + 4
. . . | . . . \                     She will row to point P   
. . . | . . . . \                        then walk to V.
. . . * - - - - - * - - - - - - *
. . . A . . x . . P . . 6-x . . V

Let \(\displaystyle AP\,=\,x\), then \(\displaystyle PV\,=\,6-x\)

The distance she will row is \(\displaystyle JP.\)
. . In right triangle \(\displaystyle JAP:\;x^2\,+2^2\:=\:JP^2\;\;\Rightarrow\;\;JP\,=\,\sqrt{x^2\,+\,4}\) miles.
. . At 2 mph, it will take her: \(\displaystyle \frac{\sqrt{x^2+4}}{2}\) hours.

The distance she will walk is \(\displaystyle PV = 6\,-\,x\) miles.
. . At 5 mph, it will take her: \(\displaystyle \frac{6\,-\,x}{5}\) hours.

Her total time for the trip is: .\(\displaystyle \L T\:=\:\frac{\sqrt{x^2+4}}{2}\,+\,\frac{6\,-\,x}{5}\) hours.

And that is the function we must minimize . . .

 

[legacyofpiracy]

Wow I cannot say how much this helped me . Thank you so much, I really appreciate your help!