Jane is 2 miles offshore in a boat and wishes to reach a coastal 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

Jane is two miles away (in point B) from point A at the beach and the horizontal distance between Jane and the village( Point C ) is 6 miles, she will leave the boat somewhere (Point D, at x distance from A) between A and C.

Then distances for the travel of Jane are:

L₁ distance to row L₁² = 2² + x² L₁ = √ 4 + x²

And the distance to walk L₂ = 6 - x

The relation between distances, time, and speed is

d = v*t then t = d/v

According to that the time for rowing will be

t₁ = ( √4 + x² ) / 2 and for walking t₂ = ( 6 - x ) /5

T = t₁ + t₂

T(x) = ( √4 + x² ) / 2 + ( 6 - x ) /5

Tacking derivatives on both sides of the equation

T´(x) = 1/2 [ ( 2*x / 2√ ( 4 + x² ) ] -1/5

T´(x) = ( 1/2 )* x / √ ( 4 + x² ) - 1/5

T´(x) = 0

( 1/2 )* x / √ ( 4 + x² ) - 1/5 = 0

5*x - 2* √ ( 4 + x² ) = 0

5*x = 2 *√ ( 4 + x² )

Squaring

25*x² = 4* ( 4 + x² )

25*x² = 16 + 4*x²

21*x² = 16

x = √ 16/21

x = 0,87 miles