Suppose that an airplane is asked to stay in a holding...

Suppose that an airplane is asked to stay in a holding pattern near an airport. The distance of the plane to the airport is described by the following. $d(t) = 142 + 73 \sin\left(\frac{2\pi}{13}t\right)$ In this equation, $d(t)$ is the distance of the plane to the airport (in miles), and $t$ is the time (in minutes) after the plane enters the holding pattern. During the first 13 minutes after the plane enters the holding pattern, when will the plane be 180 miles from the airport? Do not round any intermediate computations, and round your answer(s) to the nearest hundredth of a minute. (If there is more than one answer, enter additional answers with the "or" button.) t = minutes or

Transcript

00:01 Hello there.

00:01 I hope this problem isn't stressing you out too much.

00:04 Let's get into it.

00:06 So here we've been given the function that represents the distance from the airport in miles, given time t, of an airplane who's been put into a waiting pattern out by the airport.

00:19 And it says here that in the first 12 minutes, when will the plane be 110 miles from the airport? so a key here, 12, that's telling us that the period is 12.

00:34 That's why they want the first 12 minutes.

00:36 We can see here that if we take 2 pi, which is the standard period for the sign function, and divide it by this value out in front of t here, 2 pi over 2 pi over 12 comes out to be 12 minutes.

00:52 So we can see here why they're asking us what time.

00:57 In the first 12 minutes it's going to be asking us about because they want us to just look between the first rotation of this pattern.

01:07 So let's take a look.

01:08 Let's figure out what's going on here.

01:10 We're told that the distance is 110 miles.

01:13 So let's plug all this in.

01:18 I'm going to reduce 2 pi over 12 to pi over 6 here just to make multiplying a little bit easier.

01:26 And let's go ahead and isolate this sign function.

01:29 So when we subtract 45 from both sides, we get negative 35, and then we divide both sides by 67 to get our ratio.

01:38 Now remember that sign just represents the ratio of sides in a right triangle given a specific angle.

01:48 So if we have the ratio and we want to find the angle, we're going to have to use the arc sign.

01:54 So i'm going to take the arc sign of negative 35 over 6.

02:00 And if i take the arc sign of sign, it leaves me with just what's left in as the arguments on the inside.

02:09 So before we multiply by 6 and divide by pi to get t, we need to double check what arc sign is going to return.

02:17 Because here we're being told that the sign value, the answer to the sign function here, is negative.

02:23 And we know from experience with the unit circle that the sign function is positive in the first and second quarter and it is negative in the third and fourth quarter.

02:39 So we are going to be looking for angles in the third and fourth quarter and the calculator doesn't give that to us.

02:47 The calculator gives us a reference angle to help us with.

02:50 So when we do the arc sign or the inverse sign, if you're using a textbook that uses this notation for the arc sign, of negative 35 over 67.

03:07 Also make sure you're in a radian mode, especially if you're in a class where it's going back and forth depending on your application.

03:13 All right.

03:14 So i'm going to round this to four decimal places here.

03:16 What we're getting here is negative 0 .5496...

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