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.
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To find the rate at which the cars are separating after 30 minutes, we need to determine the distance between the two cars after 30 minutes and then calculate the rate of change of that distance.
Let's break down the problem step by step:
1. Calculate the distance each car has traveled after 30 minutes.
- Car X: Since it is traveling East at a speed of 50 km/hr, its distance can be calculated using the formula: Distance = Speed × Time. So after 30 minutes (which is 0.5 hours), Car X would have traveled a distance of 50 km/hr × 0.5 hr = 25 km.
- Car Y: Since it is traveling South at a speed of 60 km/hr, its distance can also be calculated using the formula: Distance = Speed × Time. After 30 minutes, Car Y would have traveled a distance of 60 km/hr × 0.5 hr = 30 km.
2. Calculate the distance between the two cars after 30 minutes.
- We can use the Pythagorean theorem to find the distance between two points in a two-dimensional plane. In this case, the two points represent the final positions of the two cars after 30 minutes. The distance between them can be calculated using the formula: Distance = √((x2 - x1)^2 + (y2 - y1)^2), where (x1, y1) and (x2, y2) are the coordinates of the two cars.
- Since Car X is traveling only in the x-axis (East) and Car Y is traveling only in the y-axis (South), their x-coordinates and y-coordinates are independent of each other after 30 minutes. So, we can simplify the formula to: Distance = √((25 - 0)^2 + (0 - 30)^2) = √(25^2 + (-30)^2) = √(625 + 900) = √1525 ≈ 39.05 km.
3. Calculate the rate at which the cars are separating.
- The rate at which the cars are separating is the rate of change of the distance between them with respect to time. Since the distance between the two cars is constant (39.05 km) after 30 minutes, the rate at which the cars are separating will be zero. This means that after 30 minutes, the cars are not getting any closer or farther apart.
So, after 30 minutes, the rate at which the cars are separating is 0 km/hr.