A challenging calculus and mechanics problem?

Question

From point A located on a highway as shown in attached figure, one has to get by car as soon as possible(in minimum time) to point B located in the field at a distance l from the highway. It is known that the car moves in the field η slower than on the highway. At what distance from point D one must turn off the highway?

http://www.flickr.com/photos/30014811@N06/2806370996/

Answer 1

Let Z denote the distance from A to D, and let y denote the distance from C to D (so the distance from A to C is (Z-y)). Let the car's speed on the highway be v1, and let the car's speed in the field be v2 (I'll let you figure out how to express that in terms of η). BC is the distance traveled in the field; since it's the hypotenuse of triangle BDC, we have

BC = √(y² + l²)

The time spent on the highway is (Z-y)/v1

The time spent in the field is √(y² + l²)/v2

The total time T, which we want to minimize, is given by

T = (Z-y)/v1 + √(y² + l²)/v2

dT/dy = -1/v1 + (1/v2) y/√(y² + l²)

Set dT/dy = 0 and solve for y. I leave the details to you.

You should consider the possibility that the solution for y exceeds Z. What then?