Related Rates Airplane Example - YouTube

In this video we will solve the following related  rates problem. An airplane traveling a constant  

speed and constant altitude of 32,000 feet  passes a tracking station as shown in the  

diagram. What is the speed of the plane in  miles per hour when the angle of elevation  

theta is 22 degrees and is decreasing  at a rate of 0.22 degrees per second?

Normally the first step we would take would be  to draw a diagram, but in this case it's been  

provided for us. But before we start writing down  relationships it's probably a good idea to take  

note of some issues involving units. We've been  asked for the speed of the plane in miles per  

hour, but we have been shown its altitude in feet. And we've been given information about the angle  

of elevation theta that is in degrees, but we  do calculus in radians. So there are going to  

be some conversions, probably, that are going to  be in order. And so for starters we should convert  

22 degrees into radians, and if you multiply  by pi over 180 you find that's about 0.3840.

And we also want to convert the negative – or,  

pardon me – the 0.22 degrees per second. It  actually is negative because it's decreasing.  

We want to convert that into some number  of radians per second, and if we multiply  

by the appropriate pi over 180 we find that  negative point two two degrees per second

is about negative point zero zero three eight four

radians per second.

But if we look even farther ahead to the  desire to have the plane of – the plane  

speed in miles per hour when we're finished, we  might even want to change this into radians per  

hour. And if we multiply by 3600 this gives  us about negative 13.82 radians per hour.  

And probably we also want to convert the 32,000  

feet into miles, and if we do that by dividing  by 5280 we get that it is about 6.061 miles.

Now we have attended to the issues of unit, and  it's appropriate to proceed to actually solving  

the problem. We're asked about this distance x and  its rate of change. We're given information about  

theta and its rate of change. We have to relate  them somehow, and since we know the height of  

this triangle the tangent seems appropriate.  So we could say tangent of theta is 6.061  

over x, and we need to treat both theta and x to  be functions of time, which of course they are.  

And so if we differentiate using the Chain  Rule, we get that the secant squared of theta  

times dθ/dt is equal to negative  6.061 over x squared, times  

dx/dt. And about the left-hand side and the  right-hand side we've used the Chain Rule.  

Here, theta of t is plugged into tangent, and so  we have the derivative of the tangent evaluated  

at theta, and we've got to multiply by dθ/dt.  Similarly over on the right side. There's a  

temptation sometimes to take expressions of  this kind and treat them as quotients, but  

it's a little more convenient if you treat them  as powers. This is 6.061 x to the negative one.  

Its derivative is going to be negative 6.061 x to  the negative 2, and then of course times dx/dt.  

And now it's appropriate to begin plugging  in. So we have the secant squared of theta,  

which at this instant is 0.3840 in radians,  times dθ/dt which is negative 13.82

radians per hour. And then on the  other side we have the negative 6.061

over something squared, times dx/dt. And  it's dx/dt at the moment when theta is  

22 degrees or 0.384 radians. I'll just say 22  degrees for convenience. And we need to know then  

what x is at the moment when theta is 22 degrees.  

Well, it wouldn't be appropriate to be plugging  these things in before we differentiated,  

but now we can go take a look at this and say,  "Well ,all right, what if that were 22 degrees?"  

Then we would be able to say  that the tangent of 22 degrees –

Or if you like, 0.3840 radians. It's  the same number – is equal to 6.061  

over x. And so x at this instant is just going  to be 6.061 over the tangent of 22 degrees,  

and I will tell you that that is very close to  15. So we'll plug that in there. Now it's just a  

matter of busting out some algebra,. And so secant  squared of course is 1 over the cosine squared,  

so we would perform that calculation, times a  negative 13.82. If we just multiply both sides,  

then, by 15 squared and divide by negative  6.06 we are going to get that dx/dt  

when theta is 22 degrees –  4.3840 radians – is about 597

miles per hour. And we've already put some energy  into dealing with the units. But if you wanted to  

double check you could go back through and check  for each of these quantities what the units  

would be, and you find that they're going to work  out. That concludes the solution of our problem.