SOLUTION: Academic Calculus- Pythagorean Theorem and Related Rates Word Problem: A plane takes off from airport A and flies east at 480 mph. At the moment of takeoff, a second plane, appr

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Question 562747: Academic Calculus- Pythagorean Theorem and Related Rates Word Problem:
A plane takes off from airport A and flies east at 480 mph. At the moment of takeoff, a second plane, approaching A from the north at 320 mph, is still 240 miles from A. Fifteen minutes later is the distance between the planes increasing or decreasing, and how fast?
Answer by issacodegard(60) About Me  (Show Source):
You can put this solution on YOUR website!
Hi, let A=(0,0). Then, the second plane starts at (0,240). Let x(t) be the distance from A to plane 1 at time t, y(t) be the distance from A to plane 2 at time 2, and z(t) be the distance between the planes at time t (I say distance because I know that plane two doesn't reach the airport in 15 minutes). We know,
x%5E2%2By%5E2=z%5E2
%28dx%29%2F%28dt%29=480
%28dy%29%2F%28dt%29=-320
So by implicit differentiation,
2x%28%28dx%29%2F%28dt%29%29%2B2y%28%28dy%29%2F%28dt%29%29=2z%28%28dz%29%2F%28dt%29%29
So at time t=1/4 hr,
x=480%281%2F4%29=120
y=240-320%281%2F4%29=160
z=sqrt%28120%5E2%2B160%5E2%29=200
So, we get
%28dz%29%2F%28dt%29=%281%2F200%29%28120%2A480%2B160%2A%28-320%29%29=32
This is positive, so the distance between them is increasing, so the planes are getting further away at a rate of 32mph.