An air traffic controller spots two airplanes at the same altitude converging to

Question

An air traffic controller spots two airplanes at the same altitude converging to a point as they fly at right angles to each other. One airplane is 150 miles from the point and has a speed of 900 miles per hour. The other is 200 miles from the point and has a speed of 1200 miles per hour.

An air traffic controller spots two airplanes at the same altitude converging to a point as they fly at right angles to each other. One airplane is 150 miles from the point and has a speed of 900 miles per hour. The other is 200 miles from the point and has a speed of 1200 miles per hour. (a) At what rate is the distance between the planes changing? mph (b) How much time does the controller have to get one of the airplanes on a different flight path?

Explanation / Answer

Ans-

x = 150
dx/dt = 450

y = 200
dy/dt = 600

Now before we start anything we need z. The question is asking us to find dz/dt. So let's use pythagorean's theorem.

x² + y² = z²
(150)² + (200)² = z²
22500 + 40000 = z²
62500 = z²
(62500) = z
250 = z

Now that we have z, we can take the derivative and find dz/dt.

x² + y² = z²
2x dx/dt + 2y dy/dt = 2z dz/dt
x dx/dt + y dy/dt = z dz/dt

150(450) + 200(600) = 250 (dz/dt)
67500 + 120000 = 250 (dz/dt)
187500 = 250 (dz/dt)
187500/250 = dz/dt
750 = dz/dt

B) Now to find the time it takes for them to reach other we can use a basic rate formula.

Rate = distance / time
Time = distance / rate

We'll use the times we computed when we found z and dz/dt.

Time = 200miles / 750miles per hou

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