A man in a car at location Q on a straight highway is moving with speed v. He decides to reach a point P in a field at a distance d from the highway (point M) as shown in the figure. Speed of the car in the field is half that on the highway. What should be the distance RM, so that the time taken to reach P is minimum?

Let the car turn of the highway at a distance 'x' from the pointM. So,RM=xAnd if speed of car in field isv, then time taken by the car to cover the distanceQR=QM−xon the highway,t1=QM−x2v(1)the time taken to travel the distance 'RP' in the fieldt2=√d2+x2v.. (2)So, the total time elapsed to move the car fromQtoPt=t1+t2=QM−x2v+√d2+x2vfor 't' to be minimumdtdx=01v[𕒵2+x√d2+x2]=0orx=d√22𕒵=d√3