5 seconds later, the cars are getting ____________ at m/s. (Enter your answer rounded to 3 decimal places)

Question

Two cars are moving towards an intersection. One car is  meters north of the intersection and moving towards the intersection at  m/s, while the other is  meters west of the intersection and moving away from the intersection at  m/s.

Answer 1

The first car is moving south at 10 m/s. The second car is moving west at 15ms. If the intersection is placed at the origin of an x-y axis, both velocities would be negative (down and away). The position of the first car is y=190. The position of the second car is x = -80. The distance between the cars is the hypotenuse of a right triangle:

s² = x² + y²

Taking the derivative of both sides of the equation and dividing by 2:

ss ' = xx ' + yy ' ==> s ' = (xx ' + yy ')/ s

5 seconds later, the first car is 140 m north of the intersection and the second car is 155 m west of the intersection

s at those positions = √((-155)²+140²) = 208.87 m from each other

Therefore, ds/dt = ((-155)(-15) + 140(-10))/208.87 = 4.428 m/s at that instant

The cars are moving closer at 4.428 m/s