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.
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
To minimize the time it takes to reach the hospital, the vehicle should aim to reach the road at a point that minimizes the total time for both the off-road and on-road portions of the journey.
Let's break down the journey into two parts: off-road and on-road.
1. Off-Road Distance:
The off-road distance to reach the road is given as 30 km. To calculate the time taken to cover this distance, divide the distance by the speed of the vehicle on the rough terrain:
Time_offroad = Distance_offroad / Speed_offroad
Time_offroad = 30 km / 30 kph = 1 hour
2. On-Road Distance:
The on-road distance from the nearest point on the road to the hospital is given as 60 km. To calculate the time taken to cover this distance, divide the distance by the speed of the vehicle on the road:
Time_onroad = Distance_onroad / Speed_onroad
Time_onroad = 60 km / 170 kph = 0.35 hours (rounded to 2 decimal places)
Now, let's consider the total time for the journey:
Total Time = Time_offroad + Time_onroad
Total Time = 1 hour + 0.35 hours = 1.35 hours (rounded to 2 decimal places)
Therefore, the vehicle should aim to reach the road at a point that is 30 km away from the starting point, as this minimizes the time taken to reach the hospital to 1.35 hours.