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Solve the given problems.In a modern hotel, where the elevators are directly observable from the lobby area (and a person can see from the elevators), a person in the lobby observes one of the elevators rising at the rate of $12.0 \mathrm{ft} / \mathrm{s}$. If the person was $50.0 \mathrm{ft}$ from the elevator when it left the lobby, how fast is the angle of elevation of the line of sight to the elevator increasing $10.0 \mathrm{s}$ later?

Calculus 1 / AB

Chapter 27

Differentiation of Transcendental Functions

Section 4

Applications

Derivatives

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alright in this problem, we have a man that is standing up here on a ledge and he is viewing an elevator going up a clear shaft. All right, So here's my elevator car. His standing horizontally from the elevator shaft. 20 meters. He's also 20 meters high on this balcony or ledge, and the elevator is rising at a rate of five meters per second. Okay, so horizontal line of sight is going to be here. Okay, so what are we trying to find? We're trying to find how quickly the angle of elevation is changing. Okay, so our angle of elevation, if he's looking at the elevator, be here. So Fada is going to be this angle. So how quickly is defeated? Et changing and the first instance? We want to know when the height of the elevator is at 10 meters. Okay, so we're going to look at the tangent because when the height of the elevator is 10 meters, this distance here is also 10. So we're gonna have the tangent of fado is equal to H over 20 or opposite over adjacent. Okay, so we're gonna take the derivative of both sides with respect to t So I'm gonna have seek in squared data de Sade a d t. Is equal to 1/20 a d h d t. All right, so we're gonna substitute in now. Sequent squared off. Well, what is data? Okay, so we're gonna come down here and say, Well, tangent off, Ada, at this point, exactly this point, it's going to be 10/20. Their fourth Aita is going to be negative. 0.4636 radiance. Hey, because we're looking down so we can look at this as a negative 10 from our angle of elevation. So seeking squared of negative 0.4636 do you say to DT is what we're trying to find is equal to 1/20 and we know D H D T is five. Okay, so this was going to give us 1.25. Di Fada DT is equal to 1/4 and the fate a d. T. Is equal to 0.2 radiance per second. All right, so the next thing we want to know is how quickly our anguish changing when the height is 40 meters. So we'll say that's up here. Okay, Well, when this is 40 are our opposite Here is going to be 20. Were just looking from here to here is 20 now, this is our fate. Oh, so we can go ahead straight from the derivative we already established here and just go ahead and substitute in so seek and squared off. Now we need to find the angle again. All right, so here, we're gonna have Tangent of data is equal to while it's gonna be 20/20 and data then is going to be 0.7854 radiance. So 0.7854 times de Seda d team is equal to once again. 1/20 times five. So we're gonna have to defeat. Aditi is equal to 1/4 and Di Fada DT this equal to 1/8. Radiant her second

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