Two cars start moving from the same point. One travels south at 30km/h and the other travels west at 40km/h. At what rate is the distance between the cars increasing two hours later?

Answer 1

To solve this problem, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Let's say that after two hours, the distance between the cars is represented by the hypotenuse of a right triangle, and the distances traveled by each car are the two sides of the triangle. Let's call the distance traveled by the car traveling south x km and the distance traveled by the car traveling west y km.

After two hours, the car traveling south has traveled 30 km/h * 2 h = 60 km. The car traveling west has traveled 40 km/h * 2 h = 80 km.

Using the Pythagorean theorem, the distance between the cars after two hours is:

Distance^2 = x^2 + y^2
Distance^2 = 60^2 + 80^2
Distance^2 = 3600 + 6400
Distance^2 = 10000
Distance = 100 km

Now, we want to find the rate at which the distance between the cars is increasing after two hours. To do this, we can take the derivative of the equation with respect to time (t), which gives us:

2 * Distance * (dDistance/dt) = 2x * (dx/dt) + 2y * (dy/dt)

We know that dx/dt = 30 km/h and dy/dt = 40 km/h.

Plugging in the values:

2 * 100 * (dDistance/dt) = 2 * 60 * 30 + 2 * 80 * 40
200 * (dDistance/dt) = 3600 + 6400
200 * (dDistance/dt) = 10000
(dDistance/dt) = 10000/200
(dDistance/dt) = 50 km/h

Therefore, the distance between the cars is increasing at a rate of 50 km/h after two hours.