A car travels down a highway at 45 m/s. An observer stands 250 m from the highway.

a) How fast is the distance from the observer to the car increasing when the car passes in front of the observer?

(b) How fast is the distance increasing 30 s later?

A. is 0

B is 44.247

To solve this problem, we can use the concept of related rates.

Let's start by labeling the given information:
- The speed of the car is 45 m/s.
- The initial distance between the observer and the car is 250 m.

(a) To find how fast the distance from the observer to the car is increasing when the car passes in front of the observer, we need to find the rate of change of the distance with respect to time.

Let's define:
- t as the time that has elapsed since the car passed the observer (in seconds).
- d as the distance between the observer and the car at time t.

Since we know the speed of the car is 45 m/s, the rate of change of the distance with respect to time is equal to the speed of the car.

So, the rate of change of the distance is 45 m/s.

(b) To find how fast the distance is increasing 30 seconds later, we need to find the rate of change of the distance with respect to time when t = 30.

Using the same variables as before, we want to find dd/dt when t = 30.

Since the rate of change of the distance with respect to time is equal to the speed of the car, we can calculate it using the given information that the speed of the car is 45 m/s:

dd/dt = 45 m/s.

Therefore, the distance is increasing at a rate of 45 m/s 30 seconds later.

To find the rate at which the distance from the observer to the car is increasing, we can use the concept of relative motion. Let's consider the scenario when the car passes in front of the observer.

(a) To find the rate at which the distance is increasing when the car passes in front of the observer, we need to consider the relative velocity between the car and the observer. Since the observer is stationary, their velocity is 0 m/s. However, the car is moving at a velocity of 45 m/s.

The rate at which the distance from the observer to the car is increasing is given by the relative velocity. So, the rate of increase of distance is equal to the velocity of the car. Therefore, the distance from the observer to the car is increasing at a rate of 45 m/s.

(b) To find the rate at which the distance is increasing 30 seconds later, we can consider the same relative velocity between the car and the observer. However, now we need to take into account the time that has elapsed.

Since the velocity of the car remains constant at 45 m/s, the rate at which the distance is increasing remains the same. Therefore, the distance from the observer to the car will still be increasing at a rate of 45 m/s, even 30 seconds later.

In summary:
(a) The distance from the observer to the car is increasing at a rate of 45 m/s when the car passes in front of the observer.
(b) Even 30 seconds later, the distance from the observer to the car will still be increasing at a rate of 45 m/s.