an air traffic controller spots two planes at the same altitude flying towards one another. their flight paths form a right angle at point p. One plane is 150 miles from point p and is moving 450mph. the other plane is moving at 450mph but is 200 miles from point p. write the distance d between the planes as a function of time t.
To write the distance `d` between the planes as a function of time `t`, we can use the formula for finding the distance traveled by an object: distance = speed * time.
Let's start by finding the time it takes for each plane to reach point P. We'll use the formula time = distance / speed.
For the first plane, which is 150 miles away from point P and moving at 450 mph:
Time taken by the first plane = 150 miles / 450 mph = 1/3 hour = (1/3) * 60 minutes = 20 minutes.
For the second plane, which is 200 miles away from point P and also moving at 450 mph:
Time taken by the second plane = 200 miles / 450 mph = 4/9 hour = (4/9) * 60 minutes = 26.67 minutes (rounded to 2 decimal places).
Now, let's denote the time after which the planes have passed point P as `t`.
- Before the first plane reaches point P (t < 20 minutes):
The distance between the planes is the sum of their individual distances from point P.
d = (150 miles + 200 miles) = 350 miles.
- After the first plane reaches point P but before the second plane reaches point P (20 minutes ≤ t < 26.67 minutes):
The distance between the planes decreases as the second plane approaches point P. We need to subtract the distance covered by the second plane during this time.
The first plane covers a distance of (450 mph * t) and the second plane covers a distance of (450 mph * (t - 20 minutes)).
d = (450 mph * t) - (450 mph * (t - 20 minutes)).
- After both planes have passed point P (t ≥ 26.67 minutes):
The distance between the planes remains constant at the minimum separation distance after they have passed each other.
d = (200 miles - 150 miles) = 50 miles.
Therefore, the distance `d` between the planes as a function of time `t` can be expressed as follows:
d =
350 miles if t < 20 minutes,
(450 mph * t) - (450 mph * (t - 20 minutes)) if 20 minutes ≤ t < 26.67 minutes,
50 miles if t ≥ 26.67 minutes.