Periodicity of algebraic models | Mathematics III | High School Math | Khan Academy

We're told Divya is seated on a Ferris wheel at time T equals zero. The graph below shows her height H in meters T seconds after the ride starts. So at time equals zero, she looks like about two. What is this? This would be one and a half, so it looks like she's about two meters off the ground.

Then as time increases, she gets as high as it looks like this is close to 30, maybe 34 meters, and then she comes back down to looks like two meters and up to be 34 meters again. So let's read the question. The question asks us approximately how long does it take Divya to complete one revolution on the Ferris wheel?

Alright, so this is interesting. This is when she's at the bottom of the Ferris wheel. So then she gets to the top of the Ferris wheel, and then she keeps rotating until she gets back to the bottom of the Ferris wheel again. So it took her 60 seconds, and T is in terms of seconds. So it took her 60 seconds to go from the bottom to the bottom again. Another 60 seconds, she would have completed another revolution.

And so let me fill that in. It is going to take her 60 seconds, 60 seconds. And we, of course, can check our answer if we like. Let's do another one of these.

So here we have a doctor who observes the electrical activity of Finn's heart over a period of time. The electrical activity of Finn's heart is cyclical, as we hope it would be, and it peaks every 0.9 seconds. Which of the following graphs could model a situation if T stands for time in seconds and E stands for the electrical activity of Finn's heart in volts?

Well, over here looks like we peaked at zero seconds, and then here we're peaking a little bit more than one. This looks like maybe at 1.1, now maybe at 2.2 and 3.3. This looks like it's peaking a little bit more than every one second, so like maybe every 1.1 seconds, not every 0.9 seconds. So I'd rule out A.

This one is peaking; it looks like the interval between peaks is less than a second, but it looks like a good business in a second. So it looks like maybe every three-quarters of a second or maybe every four-fifths of a second, not quite nine-tenths. This first peak would be a little bit closer to one, but this one is close.

Choice C is looking good. The first peak is at zero, then the first peak; this looks pretty close to one, was less than one. It looks like a tenth less than one, so I like choice C. Now choice D looks like we're peaking every half second, so it's definitely not bad.

So this looks like a peak every 0.9 seconds. This is the best representation that I can think of, and you can actually verify that you have a peak every 0.9 seconds. You're gonna have four peaks in 3.6 seconds. So one, two, three, four. This is it; this looks like it's at 3.6. Over here, you have one, two, three, four. You've had four peaks in less than three seconds, so this definitely one—this one definitely isn't 0.9.

So instead of just forcing yourself to eyeball just between this peak and that peak, you can say, well, if for every 0.9 seconds, how long would three peaks take or four peaks? Then you can actually be a little bit more precise as you try to eyeball it. So we can check our answer and verify that we got it right.