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.

the planes' distances from p form a scaled-up 3-4-5 right triangle, so at the time specified, d, the hypotenuse, is 250

at time t hours later,

d^2 = (150-450t)^2 + (200-450t)^2

That's kind of nasty, so a simpler formula would be,

Let x be the distance of the first plane. Then the second plane is 4/3 x away from p.

d^2 = x^2 + (4/3 x)^2 = 25/9 x^2
since x = 150-250t,

d^2 = 25/9 (150-450t)^2
= 25/9 * 150^2 (1-3t)^2
= 25*2500 (1-3t)^2

To write the distance, d, between the two planes as a function of time, t, we can use the Pythagorean theorem.

Let's consider the right-angled triangle formed by the flight paths of the planes, with point P at the right angle. One plane is 150 miles away from point P, and the other plane is 200 miles away from point P.

According to the Pythagorean theorem, the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides. In this case, the distance between the two planes, d, is the hypotenuse, and the distances from the planes to point P are the other two sides.

Applying the Pythagorean theorem, we have:
d² = 150² + 200²

Simplifying the equation:
d² = 22500 + 40000
d² = 62500

Taking the square root of both sides:
d = sqrt(62500)
d = 250

Therefore, the distance, d, between the two planes is a constant value of 250 miles. It is not dependent on time, t, as both planes are moving towards each other at the same speed. So the function representing the distance, d, as a function of time, t, would be a constant value of 250 miles.