Two cars are driving towards an intersection from perpendicular directions. the first car's velocity is 222 meters per second and the second car's velocity is 999 meters per second. at a certain instant, the first car is 888 meters from the intersection and the second car is 666 meters from the intersection. what is the rate of change of the distance between the cars at that instant (in meters per second)? a -8.4 b negative square root of 85 c -10 d -7
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:
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:
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:
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.
no, because the sides are not congruent