For a better understanding of the solution provided here please find the diagram in the attached file.
In the diagram, A is the intersection. B is the position of the first car and C is the position of the second car. As can be clearly seen, as per the directions given in the question, the cars and the intersection make a right triangle.
The distance between the first car and the intersection is and the distance between the second car and the intersection is
. The distance between the two cars is depicted by
. As we can see,
is the hypotenuse of the right triangle. At the given instance the distances are 8 meters and 6 meters respectively. Thus, by the Pythagorean Theorem the hypotenuse will be:
...............(Equation 1)
Now, we know that at any instant the Pythagorean Theorem holds and so we will have, in general:
Now, implicitly differentiating the above formula with respect to time, we get:
This can be further simplified by dividing both the sides by the common factor 2 as:
.................(Equation 2)
As we can see from the questions and the diagram, and
. The negative sign is there because the distances y and x are reducing as the cars approach the intersection.
Applying this knowledge to (Equation 2) along with the fact that as per (Equation 1), , we get (Equation 2) to become:
Therefore,
Thus, the rate of change of the distance between the cars at the given instant (in meters per second) is .
Therefore, out of the given options, option D is the correct one.
