A ferris wheel is built such that the height h in feet above...

A Ferris wheel is built such that the height $ h $ (in feet) above ground of a seat on the wheel at time $ t $ (in minutes) can be modeled by $ h(t) = 53 + 50 \sin \left(\dfrac{\pi}{16} t - \dfrac{\pi}{2}\right) $. The wheel makes one revolution every $ 32 $ seconds. The ride begins when $ t = 0 $. (a) During the first $ 32 $ seconds of the ride, when will a person on the Ferris wheel be $ 53 $ feet above ground? (b) When will a person be at the top of the Ferris wheel for the first time during the ride? If the ride lasts $ 160 $ seconds, how many times will a person be at the top of the ride, and at what times?

Transcript

00:02 All right, here we have the ferris wheel problem with a given function, and the problem says that time is measured in minutes.

00:09 However, if you work it out and compare your answer with the answer in the book, you'll find that everything works in seconds, not minutes.

00:16 So i'm going to assume that that was just a typo.

00:19 Okay, so then we're doing part a, and we're figuring out when a person will be 53 feet above the ground during the first 32 seconds.

00:27 So the person's height is 53.

00:29 So we're going to set the equation equal to 53 and we'll solve for t.

00:34 And what i'm going to do is let theta equal pi over 16 times t minus pi over two.

00:42 To simplify writing things out and to simplify our work in a little while.

00:46 So let's just do that substitution.

00:48 So 53 plus 50 sine theta is equal to 53.

00:51 The person is at a height of 53.

00:53 Let's subtract 53 feet from both sides and we have 50 sine theta equal 0 and divide both sides by 50.

01:00 So we're looking for the times, or we're looking for the theta values, i should say, when the sign is zero.

01:06 So sign is zero when theta equals zero, when theta equals pi, when theta equals 2 pi, 3 pi, etc.

01:16 So now we need to incorporate t back into this.

01:20 So remember that theta is is pi over 16 t minus pi over 2.

01:26 So let's solve that for t.

01:27 So pi over 16 times t minus pi over 2 equals theta.

01:33 Add pi over 2 to both sides and multiply both sides by 16 over pi.

01:45 I left something out here.

01:47 Excuse me.

01:49 Left out my theta.

01:50 I was thinking it was a zero, i guess.

01:52 Theta plus pi over 2.

01:54 Multiply both sides by 16 over pi.

01:56 And we get 16 theta over pi plus 8.

02:00 So that's our conversion factor from theta to t.

02:03 Okay, so let's figure out the different times when we have theta equals 0, pi, 2 pi, etc.

02:11 So if theta is 0, then t is 16 times 0 divided by pi plus 8.

02:21 T is 8.

02:22 If theta is pi, then t is 16 times pi divided by pi, that's 16 plus 8, that's 24.

02:33 And if theta is 2 pi, then t will be 16 times pi.

02:37 Times 2 pi divided by pi plus 8, that will be 32 plus 8, and that will be 40.

02:44 Well, we were told to find only the times between 0 and 32, so let's scratch that one...

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