## To find the time when the two vehicles are closest to each other and the distance between them at that time, we can set up a distance-time equation for each vehicle.

Let's assume that the intersection point P is at coordinate (0,0), with the north-south highway being the y-axis, and the east-west highway being the x-axis.

Vehicle 1 is traveling east at a constant speed of 60 km/h. This means its position can be represented by the equation x = 60t, where t is the time in hours since 1 pm.

Vehicle 2 is traveling south at a constant speed of 80 km/h. This means its position can be represented by the equation y = -80t + 5, where y is the position in kilometers and t is the time in hours since 1 pm. Since vehicle 2 is initially 5 km north of point P, the constant term is added.

To find the point of closest approach, we need to find the point where the distances of the two vehicles from each other is minimized. In other words, we need to find the value of t that minimizes the distance between the two vehicles.

The distance between two points in a coordinate system is given by the formula:

d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

In this case, we want to find the value of d when x = 60t and y = -80t + 5.

Substituting these values into the distance formula, we get:

d = sqrt((60t - 0)^2 + (-80t + 5 - 0)^2)

= sqrt((60t)^2 + (-80t + 5)^2)

= sqrt(3600t^2 + 6400t^2 - 800t + 25)

To find the minimum distance, we can take the derivative of the distance function with respect to t, set it equal to zero, and solve for t.

d/dt (sqrt(3600t^2 + 6400t^2 - 800t + 25)) = 0

After differentiating and simplifying, we can solve this equation to find the value of t.

However, note that the question is about finding the time when the vehicles are closest to each other, and not specifically the minimum distance between them. The minimum distance occurs when the derivative is equal to zero, but it could be either the minimum or maximum distance.

To determine whether it is the minimum or maximum distance, we can check the second derivative. If the second derivative is positive, then it is a minimum distance. If it is negative, then it is a maximum distance.

Taking the second derivative of the distance function and evaluating it at the obtained value of t, we can determine whether it is a maximum or minimum distance.

Once we find the value of t that gives either the minimum or maximum distance, we can substitute it back into the distance equation to calculate the distance at that time.

By following these steps, we can find the time and distance at the point of closest approach between the two vehicles.