Answer:
0 hours
Explanation:
To find the rate at which the distance between the two cars is increasing, we can use the Pythagorean theorem, as the cars are moving at right angles to each other. Let's denote the distance between the cars as "d" (in miles), and after 5 hours, we want to find how fast "d" is changing.
We have the following information:
Car 1 (moving south) is traveling at 28 mi/h.
Car 2 (moving west) is traveling at 70 mi/h.
Let's set up a right triangle where:
The horizontal leg represents the distance traveled by Car 2 (70 mi/h * 5 h = 350 miles).
The vertical leg represents the distance traveled by Car 1 (28 mi/h * 5 h = 140 miles).
The hypotenuse represents the distance "d" between the two cars.
Now, we can use the Pythagorean theorem:
d² = (horizontal distance)² + (vertical distance)²
d² = 350² + 140²
d² = 122500 + 19600
d² = 142100
Now, we'll take the square root of both sides to find "d":
d = √(142100)
D ≈ 377.36
So, after 5 hours, the distance between the two cars is approximately 377.36 miles. To find the rate at which this distance is increasing, we can differentiate "d" with respect to time (t):
dd/dt = (d/5) * (d/dt)
Now, we know that dd/dt represents the rate of change of "d," and we want to find it after 5 hours. We already have "d" (377.36 miles), and d/dt represents the rate of change of "d," which we want to find.
So, let's plug in the values:
dd/dt = (377.36/5) * (d/dt)
Now, we need to find d/dt, the rate at which the distance is changing. Using the chain rule, we can find d/dt:
dd/dt = (377.36/5) * (d/dt)
dd/dt = (377.36/5) * (d/dt)
Now, d/dt represents the rate at which the distance between the cars is increasing after 5 hours, and we can calculate it as follows:
dd/dt = (377.36/5) * (d/dt)
dd/dt = (377.36/5) * (0) # Since the distance between the cars is not changing along the vertical or horizontal direction after 5 hours.
dd/dt = 0
So, after 5 hours, the rate at which the distance between the two cars is increasing is approximately 0 miles per hour.