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?
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 travel time over both the terrain and the road.
Let's break down the problem into two parts:
1. Traveling over the terrain:
- Distance: 30 km
- Speed: 30 kph
The time taken to travel the distance over the terrain can be calculated using the formula: time = distance / speed.
So, the time taken over the terrain is: 30 km / 30 kph = 1 hour.
2. Traveling on the road:
- Distance: 70 km - x km (where x is the distance from the road to the point where the vehicle aims to reach)
- Speed: 180 kph
The time taken to travel the distance on the road is: (70 km - x km) / 180 kph = (70 - x) / 180 hours.
Now, we can add the times for traveling over the terrain and on the road to get the total travel time, T:
T = 1 hour + (70 - x) / 180 hours
To minimize the total travel time, we need to find the value of x that minimizes T. This can be done by differentiating T with respect to x and setting it to zero:
d(T)/d(x) = 0
d/dx (1 + (70 - x) / 180) = 0
Now, group the terms and solve for x:
[ -1 / 180 ] = 0
Since there is no real solution, it means there is no minimum value of x. In other words, the vehicle should aim to reach the road at point x = 0 km (the starting point itself) to minimize the total travel time.
Therefore, the vehicle should aim to reach the road at the starting point (0 km) to minimize the time it takes to reach the hospital.
To minimize the time it takes to reach the hospital, the vehicle should aim to minimize the total travel time, considering both the time spent driving on the terrain and the time spent driving on the road.
Let's denote:
- x as the distance (in km) the vehicle aims to reach down the road from the point it starts from.
- d as the total distance (in km) from the starting point to the hospital.
Now, we can calculate the time spent on the terrain and the time spent on the road based on the given information.
Time spent on the terrain:
The distance traveled on the terrain can be calculated as (30 - x) km, and the speed is 30 kph. Therefore, the time spent on the terrain is (30 - x) / 30 hours.
Time spent on the road:
The distance traveled on the road can be calculated as (x + 70) km, and the speed is 180 kph. Therefore, the time spent on the road is (x + 70) / 180 hours.
Total travel time:
Adding the time spent on the terrain and the time spent on the road gives the total travel time: (30 - x) / 30 + (x + 70) / 180 hours.
To find the value of x that minimizes the total travel time, we can take the derivative of the total travel time with respect to x and set it equal to zero:
d/dx [(30 - x) / 30 + (x + 70) / 180] = 0
Simplifying the above equation, we get:
-1/30 + 1/180 = 0
Now, we can solve this equation to find the value of x.
-1/30 + 1/180 = 0
-6/180 + 1/180 = 0
-5/180 = 0
5 = 0
The equation 5 = 0 is not satisfied, which means that there is no solution to the equation. Therefore, we cannot find a value of x that minimizes the total travel time.
In this case, since there is no solution, the vehicle should aim to reach the road as close to the hospital as possible in order to minimize the time it takes to reach the hospital.
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