A related rates question-question is in the details section?

Question

An aircraft is flying at a constant altitude with a constant speed of 600 miles per hour. An antiaircraft missile is fired on a straight line perpendicular to the flight path of the aircraft so that it will hit the aircraft at a point P. At the instant the aircraft is 2 miles from the impact point P, the missile is 4 miles from P and flying at 1200 miles per hor. At that instant, how rapidly is the distance between the missile and aircraft decreasing?

Answer 1

I think Nick B came up with the right answer.
I'll try to present it in simpler form.

Because the two objects are approaching the point of impact at constant speeds and in straight lines, the rate at which the distance between them is decreasing will be constant until impact.
So all we have to do is calculate the current distance between them and the time to impact, and divide to get the rate that the distance is decreasing.

Distance between aircraft and missile:
sqrt(2^2 + 4^2) = 2sqrt(5) miles

Time to impact:
4 miles / 1200 mph = (1/300) hour

Rate at which distance is decreasing:
2sqrt(5) miles / (1/300) hour = 600 sqrt(5) mph

Note: this is equivalent to Nick B's answer, but it has the square root in the numerator instead of the denominator.

Answer 2

Let x equal the distance from the airplane to P.
Let y equal the distance from the rocket to P.
Let s equal the distance between the rocket and the airplane.
We know that dx/dt = 600mph and dy/dt = 1200 mph
We want ds/dt when x=2 and y=4
By using the pythagorean theorem, x^2 + y^2 = s^2
2^2 + 4^2 = s^2 Therefore, s= the square root of 20 or 2*root(5)
Thus, we want to know ds/dt when s= 2*root(5)
Now, we differentiate the pythagorean theorem with respect to time and get
2s*ds/dt= 2x*dx/dt + 2y*dy/dt
cancel the 2's
s*ds/dt= x*dx/dt + y*dy/dt
substitute s with 2*root(5), x with 2, y with 4, dx/dt with 600, and dy/dt with 1200
2*root(5)*ds/dt=2*600+4*1200
ds/dt=3000/root(5) miles per hour

Answer 3

x^2+y^2=s^2
where s is the distance of the plane from P and y is thedistance of the missile from P
2s*ds/dt=2x*dx/dt+2y*dy/dt
2rt(4^2+2^2)ds/dt=2*2*600+2*4*1200
2rt20*ds/dt=2400+9600
ds/dt=12000/4rt5
=300rt5/5
=60rt5 miles per hour
=60*2.236=134.3 mph
the distanceis decreasing at 134.3mph

Answer 4

529p² - x² = a hundred subsequently, 529p² = x² + a hundred : p² = ( x² + a hundred) / 529 : p = (x² + a hundred)^½ / 23 we are advised the fee of replace of x a week is a fall of 800 cartons a week for that reason dx/dt = -800. And, dp/dx = (x/23) * (x² + a hundred)^(?½ ). making use of the Chain Rule we've : dp/dt = (dp/dx) * (dx/dt) whilst x = 23 then the fee at which p (the wholesale fee is changing) is :- dp/dt = (23/23) * (23² + a hundred)^(?½ ) * (-800) = -800 / ?(529 + a hundred) = -800 / 25.08 = -31.9 for that reason the wholesale fee is dropping at $31.9 in keeping with carton a week