Hard Math Problem-Help?

Question

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

Answer 1

Put Janes position as the point P(0,2)
Then put a point on the shoreline at B(x,0) and put another point V(R,0) which represents the location of the village.

The distance from her current location to the point B(x,0) is:
sqrt(x^2+2^2) = sqrt(x^2+4).
The distance from the point B (where she lands) to the village is R-x
The time rowing is sqrt(x^2+4)/2, and
the time walking is (R-x)/5

The total time then is T= sqrt(x^2+4)/2 +(R-x)/5
dT/dx = 1/2 * 2x/sqrt(x^2+4)/2 -1/5
=2x/sqrt(x^2+4) -1/5
Setting this =0 we get:
2x/sqrt(x^2+4) = 1/5
4x^2/(x^2+4)= 1/25
x^2+4 = 100 x^2
4 =99x^2
x^2=4/99
X=SQRT(4/99) = 2/SQRT(99)
So she should land her boat at 2/sqrt(99) miles down the river

Answer 2

She should row directly to shore and walk the rest of the way since she walks more than twice as fast as she rows.