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Problem 47 Hard Difficulty

A plane flying with a constant speed of $ 300 km/h $ passes over a ground radar station at an altitude of $ 1 km $ and climbs at an angle of $ 30^o. $ At what rate is the distance from the plane to the radar station increasing a minute later?

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Video Transcript

All right, We've got a question here that describes. The plane is flying at a constant speed. 300 kilometers per hour. Pass over ground radar station, altitude of one kilometer, and it climbs at an angle of 30 degrees. Wanna calculate what rate is the distance from the plane to the radar station increasing a minute later. First of all, I'm gonna go and draw out a graph. Excuse me? A figure and would have. All right, we've got a when a point, b, Let's see. And then this angle here is pi over six. And as we're told, it's 30 degrees. And then we're told that this distance is one kilometer. This will be a point p. We know our plane, Pete. All right, Now we want to figure out at what rate is the distance from the plane to the radar station increasing a minute later. Okay, so if we let the radar be appoint a and the plane this point will be actually, this will be the plane, and our radar will be here. We know that it's climbing at a rate of 30 degrees just prior or six. Then we can write out the angle a BNC ABC to be equal to I over to plus pi over six. And the reason why is because we have this angle agrees there. And that would be your You're piratey. Okay, Then we go and plug that in calculator. You will get something. We'll get to pi over three and then if we use the triangle property, we can calculate co sign of two pi three for this triangle to be the same thing as the distance from a B squared plus B C squared minus ace eastward. And we would divide that all by two times a B distance multiplied by BC distance. Okay, Now, if we let a c so a see this distance here be equal toe why we can condense some of these, uh, some of these variables And if we let b the c b equal to X. All right, let me go get different color Read. That's why X And we already know that our A b would be one way. Have one here. That one got one there. If we go ahead and plugged that in into this equation, you have a A B squared. The navy is one toe one squared, then you have a B C, which you know is X squared A. C. We've already said it's why squared and then a B is one. You have to multiply by BC, which is X right now. If you plug this into the calculator cosine of two pi over three, it's the same thing. It's negative one half. Then we can simplify this equation here. So it's in terms of why, Yeah, why swear it would be equal to negative half times two x to cancel out. And then you have a negative one negative X squared and then all of that in a negative So you would have a Y squared is equal to a positive one over X excusing just positive acts plus an X squared plus one. And we'll rewrite this so that the larger vary wasn't front. All right, now the distance climbed. After one minute, we can say the same thing as one over 60 hours. And if we plugged that in into our speed, which we're told this 300 kilometers per hour and we can say after one minute, five kilometers has been crossed. Okay, so we could say that this distance here. I'll do green this time X, it's going at 300 miles kilometers per hour. And that would mean that it's going in one minute. It would have traveled right? If you have 300 kilometers per hour, then you have one hour is equal to 16 minutes. So you would say that you traveled in one minute? Okay, One minute you traveled five kilometers and that will be equal to R X. So then your why in calculate By plugging in five kilometers for X, he would say why? Squared is the same thing as five kilometers squared plus five kilometers plus one, which would be 25 plus five plus one, which is 31. And then why would be to square root 30. Okay, now, once you get to that point, you can differentiate this equation with respect to t and plug in the values that you've been able to cabinet. Okay, so you would differentiate. Excuse me? Not this equation. But you would differentiate this equation with respect t So you would get de of y over x to Why? Excuse me? De y over t t. Why here? Y o t t. It's gonna be equal to two x DT two x t x g waas a t x t t. Right, Because if you differentiate this, you get a two x multiplied by DX DT and then you get a DX DT here and then with one just goes away. All right. Now, if we saw for D y over d X, we would have a two x plus one suffered ugtt. It goes to two X plus one makes you factor If you simplify this into one term and you have to divide it all by two y because you have to wire all right. And so now finally, you plug in all the values that you have, you have X equals 25 You have to multiplied by five plus one two multiplied by why which we know is square with 31 and then our change in X over changing t waas gonna be 300. That's what we're given. And if you plugged that into your Andy dandy calculator, you should get a value that is equal to 296 kilometers approximately to 96 kilometers. Our all right, so we can confirm that the rate that the distance from the plane to the radar station increasing a minute later was 296 kilometers per hour. All right, well, I hope that clarifies the question. Thank you so much for watching.

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Video Transcript

James Stewart

Calculus: Early Transcendentals

Anna Marie Vagnozzi

Heather Zimmers

Kristen Karbon

Samuel Hannah

This textbook answer is only visible when subscribed! Please subscribe to view the answer

Calculus: Early Transcendentals

Anna Marie Vagnozzi

Heather Zimmers

Kristen Karbon

Samuel Hannah

James StewartCalculus: Early Transcendentals

Anna Marie Vagnozzi

Heather Zimmers

Kristen Karbon

Samuel Hannah

James Stewart

Calculus: Early Transcendentals