Time Optimization?

Question

Jane is 2 mi offshore in a boat and wishes to reach a coastal village 6 mi down a straight shoreline fromt eh point nearest the boat. She can row 2 mph and can walk 5 mph. where shoudl she land her boat to reach the village in the least amount of time?

Answer 1

This problem comes up in real life, the real life of dogs. If a dog wants to retrieve a stick from the water, it runs along the beach parallel to the water for some distance, and then swims diagonally out to the stick. The dog runs faster than it swims. How far should it run along the beach? Dogs know.

In Jane's case, let the point of landing be n "fromt he point nearest the boat," and the amount of walking is 6 - n.

The rowing distance is the hypotenuse of a triangle with sides 2 and n.

The rowing distance is thus

r = sqrt[4 + n^2]

The time taken is r/2 + (6 - n)/5.

Take the first derivative of this with respect to n and set it equal to zero.

0 = (1/2)dr/dn - 1/5

Now, dr/dn = (1/2)(4 + n^2)^(-1/2) * 2n.

So,

0 = (n/2)(4 + n^2)^(-1/2) - 1/5