INSTRUCTOR: A Ferris wheel is 60 feet in diameter. It makes one revolution every 100 seconds. You climb up 6 feet of stairs to get on the wheel at its lowest point. Model the height of a rider as a sinusoidal function and graph one revolution. OK. So I like to start with a sketch of what's going on here. So you have a Ferris wheel, looks something like that. And usually, a Ferris wheel, it has these sort of support legs, like that, and they're on the ground. And so here, we'll draw the ground here like that. 

And we're told that you have to walk up 6 feet of stairs to enter the ride. So you have this step here, and then you get on the ride here. Now, the height of these stairs, they tell us, are 6 feet. We also know the diameter of the wheel is 60 feet. So from the very top of the wheel to the very bottom of the wheel, like this, that's 60 feet. That's the diameter. Obviously, this isn't to scale, but that's OK. 

We also know that it makes one revolution every 100 seconds, so we're going to keep that in mind. And now we're going to try to model the height of a person on the wheel, starting at the bottom as you go around, using an equation. Now, I like to make the graph first. They ask you to graph one revolution after, but I find it easier to make the graph and then make the equation. So let's go ahead. We'll make our axes like this. This will be t. This will be h at t, your height at a particular time in seconds. 

So when you enter the ride, you get on at a height of 6. So you start off at a height of 6. And your maximum height is 60 feet above where you enter. Well, you're already 6 feet off the ground. So 6 plus 60 means you're 66 off the ground at the very top. So that's your maximum height. If you want to know where you're at in the middle, the equation of your axis, which you should know-- you should label-- it's going to be halfway between 6 and 66. 

Well, think about it this way. The radius of the circle, that distance is 30. So when you're in the middle, you're 36 feet off the ground. So this is at 36. And sure enough, 36 is halfway between 6 and 66. It's 30 more than 6 and 30 less than 66, so it is right in the middle. So it makes sense. What's also nice about this process is everything kind of lines up. Like, your minimum value lines up with your minimum value. The ground lines up with 0. The axle of the wheel lines up with the axis of the graph. And then the maximum point lines up with the maximum point on the graph. Everything lines up nicely. 

Now, with that in mind, we can go ahead and graph this. So you start at a minimum value of 6. And then you start going around the circle, and you start going up. And you keep going up, and you get to the maximum value, and you start coming back down. And then you come back down to your minimum height all over again. 

Now, this graph here represents your height as you go around the circle. And it makes sense because these are going up. You're here. Then you get higher and higher and higher and higher. Then you get to the middle. And then you keep going up to the top, and then you start slowing down as you get to the top. Like, right here, you're going up pretty fast right. But over here, you're going up pretty slow. You're slowing down. 

You can see that in the graph, right? You start slow, then you get fast, right? It's going up very quickly, and then it kind of slows down. And then it gets fast again, and it starts slowing down. Again, you're not actually moving faster or slower. It's just that your height, your change in height, is changing more quickly or more slowly. You're not going faster or slower. Just the height is changing more slowly. Because for this section here, you're moving a lot horizontally but not much vertically. 

And then in this section here, you're moving a lot vertically but not much horizontally. This only measures the change in your vertical direction in your height. OK. So now the last thing we have to label on this graph is the period. Well, we know you make one revolution every 100 seconds. So at this point, we must be at 100 seconds, meaning that at your maximum, you're going to be at 50 seconds. And when you're at your averages, that 36-foot mark, you're going to be at 25 seconds and 75 seconds. 

I'm just dividing 100 into four sections here-- so 0 to 25, 25 to 50, 50 to 75, 75 to 100. Pretty self-explanatory, right? OK. Now we have to actually come up with the equation for this. So I'm going to say h of t is equal to-- now, when we write this out, we have to make a decision. Do I want to use sine, or do I want to use cosine for this function? Well, it kind of looks like an upside down cosine graph, right? Because remember, cosine usually goes like that. It starts at the top, goes down, comes back up. So I think we should use cos. So I'm going to use cos. 

What would my amplitude be? Because the amplitude is the thing that goes on the front. Well, the amplitude is the distance from the middle to the top or the middle to the bottom. Well, that's just your radius in this case. It's the distance from your middle to the top or middle to the bottom of the circle of the wheel. In this case, your radius is 30. We talked about that. Now, if you forgot, you could measure this distance from 6 to 36 or 36 to 66. Those are both equal to 30. So of course, your amplitude for this one is just going to be 30. So you put 30 on the front. 

And now remember what we said, that this is the upside-down version of the cosine graph. So it's been vertically reflected. So we need to make that negative 30. Now, what else do we know? Well, we know that the axis right here is at a height of 36. So I can say plus 36 on the end of this equation. I know I have t. And that's not being shifted left or right because it's just an upside-down cosine graph. There's no phase shift. We're not moving to the left or to the right. 

The only thing that I need to find is this k value here in front of the t to change the period. So I have to recall, what's that relationship? Well, k is equal to 360 degrees divided by the period. Or you could say the period's equal to 360 divided by k. Those two are interchangeable. So how do I find the k? I plug in my period, which is 100 seconds, so 360 divided by 100, which is equal to 3.6. So 3.6 is what I'm going to plug in right here, 3.6. You can put that in a bracket if you want. So it's 3.6t, is what I've written there. 

This is the equation or the function that represents this scenario where you're going around the Ferris wheel. Now let's say they ask you some follow-up questions. What if they asked you, how high are you at a particular time? Like, if they said at 25 seconds, what's your height? Well, you can look at the graph and see, OK, 25 seconds, you go up. You're at 36 feet. But they could ask you something more specific. They could say, at 27 seconds, how high are you? 

Well, you would go, and you'd realize, OK, I'm a little bit above 36 feet. But how high is that, exactly? Well, all you would have to do is plug in 27 into your formula. Find h at 27 and calculate it. We can actually do that. If I want to find the height at 27 seconds, I just plug in 27 for t. So you'd have minus 30 times cos of 3.6 times 27 plus 36. And then you would just go ahead, and you would calculate this. 

And when you do that, you get an answer of 39.76 feet, approximately, if you round your answer. So 27 seconds, you're at 39 feet. And that makes sense with the graph, right? 27 seconds would be about here, a little bit after 25. And that would put you at a height a little bit bigger than 36, like the one we found, 39.76. So that makes sense. 

Now, they could also ask you a question like, at what times are you at a height of 40 feet? So you look at the graph here, and you say, OK, well, 40 feet's about here. And then you can look, and you can say, OK, where's this point and this point? Because that's also at that same height. And then you look down and you see, OK, here and here. What are those times? How do you find that? Well, to find those, all you have to do is plug in 40 for your h and solve for t. So 40 equals negative 30 cos 3.6t plus 36. 

Now let's try to solve for t. You move 36 to the other side. So 40 minus 36 is 4. And you want to divide by 30, divide both sides by 30. So you'll have negative 4/30 is equal to cos of 3.6t. And then I need to get rid of cos. So you're going to take the cos inverse of both sides cos inverse of 4 over 30 equals 3.6t. If you do cos inverse of 4/30, you get 97.66, approximately. That's equal to 3.6t. 

Divide both sides by 3.6, and you're going to get t is equal to 27.13 seconds. Now, 27.13, if you look at the graph up here, it makes sense, right, this time here? 27.13. That makes sense, right? It's right after 25, just a little bit to the right, kind of similar to the question we just had, similar numbers, anyway. But that's just one time. That gives you the time when you're at this height right here. 

But you're at the same height over here. How can I find that time? Well, thankfully, these functions are symmetrical, and they're cyclical. They repeat over and over again. So we can find out what this one is if we know what this one is because this is the same distance from the start as this time will be from the end. 

So we can actually determine that, well, this distance is 27.13. This distance here is also 27.13. So I can subtract that from 100 seconds. So I can say 100 seconds minus 27.13 seconds, and that's going to give me 72.87 seconds. So that's the other time when we're at this height. And if you look at the graph, it makes sense. Right here is 72.87 because, look, it's right before 75 seconds. So it makes sense. The answer here makes sense. 

OK, let's do another example. Let's say you're told that the hottest day of the year is June 7. And on June 7, it's 29 degrees Celsius. And let's say we're told that the coldest day of the year is-- oh, we don't know what it is, actually. We're not told that. But we know that it's 14 degrees Celsius on that day. All right. And this is on average at a particular time. 

Using a trig function or a sinusoidal function, model the temperature using 365 days. So we got to model the average temperature every day for a year. And then they're going to ask you, how many days after June 7 is the first spring day when the temperature reaches 20 degrees Celsius? So that's an additional point. We'll get to that later. So for now, let's just try to model this. 

So you have a function. We're going to set up our axes like this. Over here, we'll have the temperature in terms of the day. And on this axis, I'll have d for days. But that's going to be days after June 7. We're going to start on June 7 at the maximum day up here at 29 degrees Celsius. And then down here, we'll have 14 degrees Celsius. That's the coldest day. So we start at the top. And this is just going to follow that same sinusoidal pattern. It repeats. So it's like a cycle. It goes up and down and up and down, up and down, keeps repeating itself. 

Now, there's lots of things here we can find. We can find the axis or the midline. It's 29 plus 14 divided by 2. It's the average between those. If you do 29 plus 14 divided by 2, that's equal to 43/2 or 21.5. So 21.5 is going to be the average. We can label this here, day 0, as being June 7. And then when we get to that point again up here, that'll be day 365, which we know is also June 6, will be the day before, because then it just repeats the entire cycle over and over and over. 

And so now we can figure out what our equation looks like. We can say t at d is equal to-- well, we know the amplitude, right? The amplitude is this distance or this distance. Distance from 14 to 21.5 is 7 and 1/2, right? From 21.5 to 29 is also 7 and 1/2. So the amplitude is 7.5. 

Now, it's a cos function. You can see that pretty clearly. So we'll put cos. I want to put some k value in front of d, but then I'll put 21.5 because that's the equation of the axis. That's your average here. That's your midline. So what value for k can I put? Well, k, as we know, is 360 divided by the period. Well, the period's 365. 

So that actually reduces to 72/73. So I'll put that number in here, 72/73. That simply converts whatever number of days we put in here into a degree that the function cos can understand and interpret. So that's the function. So that's part of the question already done. 

The other part of the question-- I'll read this again-- is, how many days after June 7 is the first spring day when temperatures reach 20 degrees Celsius? And they say first spring day because you obviously reach 20 degrees Celsius here, but at that point, it's still getting colder. And as it gets colder, you get to the winter. 

Now, as you come out of the winter, here in spring, you get to 20 degrees Celsius again. I'll say this is 20 degrees Celsius. So you reach 20 degrees Celsius twice-- while the temperature is dropping into the winter and while it's recovering into the spring and into the summer. So I don't want to find this time. I want to find this time here. How many days is this after June 7? Well, how can we find that? We set the temperature to 20, and we solve for d. So let's go ahead and do that. 

We'll have 20 is equal to 7.5 cos 72/73 d plus 21.5. OK, let's solve for d. I'm going to move the 21.5 to the left. 20 minus 21.5 is negative 1.5. Divide both sides by 7 and 1/2. So divide by 7.5. That's equal to cos of 72/73, just like that. 

I can now take the cos inverse of both sides, cos inverse of negative 1.5/7.5. Cos inverse on the right is just going to cancel. I'll get 72/73 d. When you cos inverse this fraction, you're going to get 101.54. That's equal to 72/73 d. 

And from there, I want to get rid of that 72/73. So I'll multiply both sides by 73/72-- again, inverse operations, so those will cancel. Those cancel. I'm left with d. d, you're going to multiply all this. d is going to equal 102.95. 

So 102.95 days-- if we look at our drawing up here, this whole year's 365. So half the year-- 365 divided by 2-- is going to be about 182.5, is halfway. And so if that's halfway and we just got 102 as our answer, that must be talking about this one here. Well, we said we didn't want that, this 102. We didn't want that because that's as it goes into the winter. We want the answer in the spring over here. 

So I have to find out what day this is. Well, just like last time, this distance to the 102 is the same distance as from here to the end because it's symmetrical. So I can do 365 minus that 102 to figure out what day we're talking about. So I'm going to do 365 minus 102.95. And I get it as an answer, 262.05. 

Therefore, it's going to be 262 days after June 7 that we reach 20 degrees Celsius in the spring. That's how you would write your final answer.