A jet flying at 450 mi/hr and traveling in a straight line at ...

Question

A jet flying at 450 mi/hr and traveling in a straight line at a constant elevation of 500 ft passes directly over a spectator at an air show. How quickly is the angle of elevation (between the ground and the line from the spectator to the jet) changing 2 seconds later?

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So, first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A drone is moving horizontally at a constant speed of 200 m per minute, at an altitude of 300 m. It flies directly above an observer standing on the ground. Determine the rate at which the angle of elevation between the ground and the line from the observer to the drone. is changing 4 seconds later. Awesome. So it appears for this particular problem, we're trying to figure out the rate at which this angle of elevation is changing 4 seconds later. So that's what we're ultimately trying to solve for is what is this rate. Of, so we're trying to determine the rate at which the angle of elevation is changing. So what is this rate of change? Awesome. So now that we know what we're ultimately trying to solve for, let's read off our multiple choice answers to see what our final answer might be, noting that they're all going to be in the same units of radiances per minute. So A is 675 divided by 2029. B is negative 675 divided by 2029. C is 1350 divided by 2029. And D is negative 1350 divided by 2029. Awesome. So, for this particular problem, first off, what we need to do is to help us better visualize what's happening. Let's create a diagram. So, I've gone ahead and put together a digital diagram for us to refer to. As you can see, we have our drone flying up above in the atmosphere at an altitude of 300 m, and then we could draw in a little stick figure, and I'm gonna draw a stick figure in blue, and we have an observer standing at the bottom on the ground, and it's looking up at our drone. And as you can see, it forms a like a hypotenuse, so it's basically essentially looking at it as if it was a triangle. So we have one of the legs of our triangle being the altitude, which is 300 m, and then we have our hypotenuse being the direct line of vision from our observer looking up at the drone, and then our distance from where the observer is standing out to where the drone is is some value X. Awesome or some distance X away. So the observer is some distance X away from the drought. Awesome. So with that said, our next step now to help us solve for this is we need to define all of our known variables, and we're trying to also figure out and determine what our target variable is. So we know the altitude of the drone, which we'll call H and H is equal to 300 m. We also know the horizontal speed of the drone, which we'll call the speed V, and the speed V is equal to 200. Meters Per minute. Awesome. And the time after which the rate of change is to be determined is T equals 4 seconds. Fantastic. So now we need to convert our time to minutes. So you can use dimensional analysis to do this conversion, or you can recall that T is equal to 4 divided by 60, which is equal to 1/15 of a minute. Fantastic. And let us note that X will be the horizontal distance from the observer to the drone, as we discussed, and theta will be the angle of elevation, and that is the value that we're going to need. To determined to help us figure out our final answer. So we need to note once again, or when I say once again, but once again we're trying to ultimately we're trying to figure out the rate of change for this angle of elevation after 4 seconds. So in order to do that, our target variable will have to be D theta by DT. So we need to Focus our attention on D theta by DT. So now, our next step is we need to write the relationship between the variables. So we need to note and recall that the horizontal distance X. After T will be X is equal to V multiplied by T. and when we recall and use trigonometric identities, we could say that tangent of theta is equal to H divided by X. And we need to note and recall that X is a function of T and this value of H will be constant. So now we need to differentiate the equation with respect to time T. So we need to take D by DT where D by DT is a fancy way of saying we're taking the derivative with respect to T, and we need to take D by DT of tangent of theta, which is equal to D by DT. Of H of X, and in order to perform this differentiation, we need to recall and use the chain rule. So we need to note that C can't square of theta. Multiplied by D theta by DT. is equal to negative H divided by X2 multiplied by DX by DT, fantastic. And now we need to substitute in the known values back into our differentiated equation. So when we do that, we will find that X is equal to 200 multiplied by 1 divided by 15, which is equal to 40 divided by 3. Meters And we need to note that. DX by DT, so when you take DX by DT which is gonna be equal to 200 m per minute. And we need to note once again that tangent of theta is going to be equal to 300. Divided by 40 divided by 3, which is going to be equal to 300 multiplied by 3 divided by 40, which will be equal to, when we simplify, 45 divided by 2. Fantastic. So now our next step is we need to take C can squared. Of theta which is equal to 1 plus tangent squared. Of theta, which is going to be equal to 1 plus when we plug in our value of 45 divided by 2 N for theta, we'll get 45 divided by 2 squared, fantastic. And now, our next step is we need to determine the final equation. We need to use the final equation to help us find the theta by DT. And when we do that, we will find that 1 plus. 45 divided by 2 squared. Multiplied by D theta by DT is going to be equal to -300. Divided by 40, divided by 3 squared, and then we need to multiply this by 200. And now what we need to do is we need to simplify, so we need to take 1 + 2,025 divided by 4 multiplied by D theta by DT, which is going to be equal to -660000. Divided by 1600 divided by 9. Fantastic. And then we need to keep simplifying, so we need to take 2,029 divided by 4 multiplied by D theta by DT, which is equal to -675 divided by 2. And finally, if we simplify. Just a few more times, we'll get that D theta by DT is equal to negative. 675. So 675 divided by 2, divided by 2029 divided by 4, which will be equal to negative, let me simplify one last time, 1350 divided by 2029, and do not forget that its units are going to be in radiance per minute. And that's it. We've solved for our final answer. Hooray, we did it. So therefore. Looking at our multiple choice answers our final answer without a doubt has to be multiple choice answer letter D, which once again is -1350 divided by 2,029 radiances per minute. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

A jet flying at 450 mi/hr and traveling in a straight line at a constant elevation of 500 ft passes directly over a spectator at an air show. How quickly is the angle of elevation (between the ground and the line from the spectator to the jet) changing 2 seconds later?

Key Concepts

Related Rates

Trigonometric Functions

Differentiation

Watch next