Final answer:
To find when two vehicles traveling towards an intersection from perpendicular directions will be closest, we calculate the minimum point of the Pythagorean distance function by differentiating and solving for time. The minimum distance between the vehicles can then be determined by substituting this time back into the distance function.
Step-by-step explanation:
To determine the time at which two vehicles traveling towards an intersection, one from the east and the other from the north, will be closest to each other, we can treat this as a problem involving the relative distance between two points moving along perpendicular paths. In this scenario, the truck is traveling east at 80 km/h, and after a certain time t, it will have traveled 80t kilometers. Similarly, a car is traveling north at 50 km/h, and after the same time t, it will have traveled 50t kilometers. Initially, they are separated by 32 kilometers along the east-west direction, and we are looking for the minimum distance between them.
Let's set up a coordinate system where the truck's initial position is at the origin, and the car is 32 km along the positive x-axis. As time progresses, the truck moves east, and the car moves north, thus their paths form the sides of a right-angled triangle.
The distance between the two vehicles at any time t can be given by the Pythagorean theorem:
To find when they are closest to each other, we need to minimize the distance function D(t). This involves finding the derivative of the distance function with respect to time t, and setting it to zero to solve for t.
After solving, we get the time t when the distance is minimized. At that point, we can then calculate the minimum distance by plugging t back into the distance function.