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A plane flies horizontally at an altitude of $ 5 km $ and passes directly over a tracking telescope on the ground. When the angle of elevation is $ \pi/3, $ this angle is decreasing at a rate of $ \pi/6 rad/min. $ How fast is the plane traveling at that time?

$\frac{10}{9} \pi \mathrm{km} / \mathrm{min}[\approx 130 \mathrm{mi} / \mathrm{h}]$

01:49

Wen Z.

00:57

Amrita B.

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 9

Related Rates

Derivatives

Differentiation

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All right, We've got a question here that describes the plane that was flying at altitude of five kilometers. We go and roll out of figure first and we've got an angle data here in the ground that is X distance. We're told that the angle of elevation is pie or three, and the angle is decreasing at a rate of pie or six radiance per minute. We want to use this information to calculate how fast the plane is traveling at that time. Okay, so first of all, we can write out an equation that that includes data and X will say tangent data is the same thing as opposite over adjacent, which is five or Rex. We'll move everything to one side and just keep X here we have X equals to five over tangent data, which can also be written at five co Tangent X. Okay, If we went ahead and took the derivative of that, then we could find the change of X over time. And that will be equal to five five times. Uh, times d over DT. And then we have and two x. So we know the derivative Contention X is the same thing as as called seek unt. Excuse me? It's the same thing as negative co ck in squared X. We have negative five now C can't squared X that all equal to t x solidity. And we're gonna have a deep data over DT on this side. Okay, now that we have that, we can go ahead and solve for our change in X all the time by plugging in our data, which we know is given as our angle of elevation, which is pirate three. And there are deep data over DT as pi over six. So you can calculate d excellent DT by just plugging in the given values. We have negative five times co Seacon Hi. Over three squared multiplied by debate over three DT which were already are told is pi over six. And then when you actually go ahead and multiply this out, you will get a pie. Excuse me, a 10 pie 59 kilometers minute. Okay. And then if you went and converted this to decimals, you would get somewhere approximately 130 miles per hour if you excuse me if you convert it thio English units. All right. And that will be your final answer there either. Or units works. You do it in s I or in English. Either one is fine. All right, well, I hope that clarifies the question. Thank you so much for watching.

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