A train is moving west along a long straight track toward a...

A train is moving west along a long straight track toward a station. An observer is located at a point 0.75 miles directly south of the train station with a camera (see figure below). Station Train 0.75 miles ? Observer (a) Let x represent the distance of the train to the station. Express x in terms of ?. x = (b) Use implicit differentiation to find dx/dt = (c) The observer notices that when the train is 2.5 miles from the station, he is having to rotate his camera at a rate of 0.06 radians per minute to keep the train in focus. Write an appropriate mathematical statement for each of these two pieces of information using the variables defined in this problem. Use Leibniz notation for any derivatives. when the train is 2.5 miles from the station means what about x? the observer rotating his camera 0.06 radians per minute means what about the angle ?? (d) How fast is the train traveling at the instant described in part (c)? Give your answer to the nearest miles per hour.

Transcript

00:01 A train is moving west along a long straight track toward a station.

00:04 An observer is located at a point 0 .7 miles directly south of the train station with a camera.

00:12 And they give us this figure.

00:14 They say, i want to let x represent the distance of the train to the station.

00:20 And i want to express x in terms of theta.

00:24 So what can i say? well, based off of the tangent ratio, i can say that the tangent of theta is equal to x over 0 .75.

00:38 Tangent is opposite over adjacent, so x over 0 .75.

00:43 Now what does that mean about x itself? well, i can multiply both sides by 0 .75.

00:49 I can say that 0 .75 times the tangent of theta is equal to x.

00:57 So my x is equal to 0 .75 times the tangent of theta.

01:06 Now in part b, they say, let's use implicit differentiation to find dx dt.

01:14 So what would dx dt be? it would be 0 .75 times the secant squared of theta times d theta dt.

01:29 So dx dt would be 0 .75 secant squared of theta d theta dt.

01:38 Now, the observer notices that when the train is 2 .5 miles from the station, he is having to rotate his camera at a rate of 0 .06 radians per minute to keep the train in focus.

01:52 We're going to write an appropriate mathematical statement for each of those pieces of information using the variables defined in the problem.

01:59 So first of all, when the train is 2 .5 miles from the station, what does that mean about x? well, at that moment, it means that x is 2 .5.

02:13 Now, the observer rotating his camera at 0 .06 radians per minute means what about the angle theta? well, that's the rate of change of theta.

02:26 So my d theta dt...

02:28 Now i do have to be careful here.

02:30 You notice theta is getting bigger or smaller this time.

02:35 It's getting smaller, right? so d theta dt is negative.

02:40 It's negative 0 .06.

02:43 And then in d, we say, well, how fast is the train traveling at the instant described in part c? we'll give our answer to the nearest mile per hour.

02:54 Well, let's see.

02:55 We already have this fact up here that dx dt was equal to 0 .75 secant squared of theta d theta dt.

03:04 Let's figure out what the secant squared of theta is.

03:08 So what can i say? well, i can say that the...

03:15 Let me do the pythagorean theorem here.

03:17 I'll make my life a little easier.

03:18 So this is 2 .5...

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