A four-wheel-drive vehicle is transporting an injured hiker to the hospital from a point that is 30 km from the nearest point on a straight road. The hospital is 60 km down that road from that nearest point. If the vehicle can drive at 30 kph over the terrain and at 170 kph on the road, how far down the road should the vehicle aim to reach the road to minimize the time it takes to reach the hospital? Rounded to 2 decimals.

Answer 1

x at 170 km/h
we want 60 -x
sqrt [ 30^2 + (60 - x )^2 ] at 30 km/h

Time = x/170 + sqrt [ 30^2 + (60 - x )^2 ] / 30
we want dT/dx = 0
0=(1/170 )+(1/30)(1/2) d/dx [ 30^2 + (60 - x )^2 ] / [ 30^2 + (60 - x )^2 ]^.5
0 = 5.88*10^-3 + (1/60) [ 2(60-x)(-1)] / [ 30^2 + (60 - x )^2 ]^.5
60 * 5.88*10^-3 [ 30^2 + (60 - x )^2 ]^.5 = 120 - 2 x
.3528 [ 900 + 3600 -120 x+ x^2] = 120 - 2 x
solve quadratic for x
answer is 60 - x