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 70 km down that road from that nearest point. If the vehicle can drive at 30 kph over the terrain and at 180 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?

Question

Steve · Answer

If the vehicle drives to a point x km from the "nearest point" then the distances are

terrain: √(30^2+x^2)
road: 70-x

So, the time is

√(900+x^2)/30 + (70-x)/180

Now just find where that has a minimum. I get x≈5km