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Finding the speed of an airplane with respect to change in angle (rate of change problem)?
I see many asked this question but no correct solution given.
Here is the one.
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An Airplane is flying at a constant speed and altitude, on a line that will take it directly over a radar station located on the ground. At the instant the airplane is 60000 feet from station, an observer notes that its angle of elevation is 30 deg and increasing at a rate of .5 deg per sec. Find the speed of the airplane.
Solution:
tan (theta) = h / x
(h is the altitude and x is the horizontal distance from airplane to the observer at
radar station any time. D will be the hypothenus, i.e., 60,000 ft.)
Take derivative to both sides,
sec ^2 (theta) d(theta)/dt = -h / x^2 dx/dt
now, at the instant, theta = 30 degree, sec^2 (30) = 4/3;
D = 60,000
h = D * sin (30) ; x = D * cos (30);
d(theta) / dt = 0.5 deg = Pi / 360
dx/dt = (4/3) * (Pi/360) * (D * cos (30))^2 / (D * sin (30))
= (4/3) * (Pi/360) * D * cos (30))^2 / sin (30)
= (4/3) * (Pi/360) * 60,000 * (3/4) / (1/2)
= 1000 * Pi / 3 ft/sec
You missed a point.
180 degree is Pi.
0.5 / 180 = x / Pi
x = 0.5 * Pi / 180 = Pi / 360. This is for the d(theta)/dt change rate.
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You already have
h = D*sin(theta) = D/2
but you again put sin(theta) in
dx/dt = -D/2 sin(theta)/sin^2(theta) d(theta)/dt
it should be
dx/dt = -D/2 * 1/sin^2(theta) d(theta)/dt
Then you will have the correct answer, 1000*Pi/3.
1 Answer
- wy125Lv 410 years agoFavorite Answer
Your solution looks good except for the following: Pi = 180 degrees not 360
Also I think it's easier to do the following:
You start with tan(theta) = h/x
Solve for x first: x = h cot(theta)
now take the derivative
dx/dt = -h csc^2(theta) d(theta)/dt
Just a bit cleaner I think but of course it doesn't change the answer.
Edit: You're right, I missed that in your solution. Now let me do the whole problem:
dx/dt = -h csc^2(theta) d(theta)/dt
h = D*sin(theta) = D/2
d(theta)/dt = 1/2 degrees/sec = PI/360 (just as you had)
dx/dt = -D/2 sin(theta)/sin^2(theta) d(theta)/dt
dx/dt = -D/2 1/sin(theta) d(theta)/dt
dx/dt = -D/2 1/(1/2) d(theta)/dt
dx/dt = -D/2 * 2 d(theta)/dt
dx/dt = -D * d(theta)/dt
dx/dt = -60,000 * PI/ 360
dx/dt = -500/3 * PI
Hmm.. I get half of what you get....
