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 concept of relative motion.
Let's consider the plane that is 150 miles from point P. Its distance from point P decreases at a rate of 450 mph (its speed) because it is flying towards point P. So, the distance of this plane from point P can be expressed as \(150 - 450t\).
Similarly, the plane that is 200 miles from point P is also moving towards point P at a speed of 450 mph. Therefore, its distance from point P is \(200 - 450t\), where \(t\) represents the time.
To find the distance between these two planes, we subtract their distances from point P:
\(d = (150 - 450t) - (200 - 450t)\)
Simplifying the expression, we get:
\(d = 150 - 450t - 200 + 450t\)
\(d = -50\) miles
So, the distance between the two planes at any given time \(t\) is constant and equal to -50 miles.