Two cars leave an intersection at the same time. Car X is travelling East at 50 km/hr and Car Y is travelling South at 60 km/hr. Find the rate at which the cars are separating after 30 minutes.

I want its complete solution

how far apart are they ?

t in hours

sqrt (x^2 + y^2)
=sqrt [(50t)^2 + (60 t)^2]
so
d = (2500 + 3600)^.5 t
d = 78.1 t
so separation speed is 78.1 mph
all the time

To find the rate at which the cars are separating after 30 minutes, we can use the concept of relative velocity.

Let's consider a right-angled triangle where the two sides represent the distances traveled by Car X and Car Y, and the hypotenuse represents the distance between the two cars.

The distance traveled by Car X in 30 minutes (0.5 hours) can be calculated using the formula: distance = speed × time. Therefore, the distance traveled by Car X is 50 km/hr × 0.5 hr = 25 km.

Similarly, the distance traveled by Car Y in 30 minutes is 60 km/hr × 0.5 hr = 30 km.

So now we have a right-angled triangle with one side measuring 25 km and another side measuring 30 km.

To find the hypotenuse, which represents the distance between the two cars, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Using the Pythagorean theorem, we can calculate the distance between the two cars:

Distance^2 = (25 km)^2 + (30 km)^2
Distance^2 = 625 km^2 + 900 km^2
Distance^2 = 1525 km^2

Taking the square root of both sides, we get:
Distance = √1525 km
Distance ≈ 39.11 km

Now, to find the rate at which the cars are separating, we differentiate the distance equation with respect to time:

d(distance) / dt = d(√(25^2 + 30^2)) / dt
d(distance) / dt = d(√(625 + 900)) / dt
d(distance) / dt = d(√1525) / dt

To differentiate, we use the chain rule:

d(distance) / dt = (1 / 2√1525) * d(1525) / dt
d(distance) / dt = (1 / 2√1525) * 0
d(distance) / dt = 0

Therefore, the rate at which the cars are separating after 30 minutes is 0 km/hr. This means that after 30 minutes, the cars are not moving away from each other or getting closer; they are maintaining a constant distance.