 Let's go to the carnal on this right triangle application here. Suppose that a rider is on a ferris wheel And the radius of that ferris wheel is a hundred and twenty five feet. It's a pretty big ferris wheel here The rider gets on at the bottom of the ferris wheel at point P not All right, and which this I should mention is 14 feet above the ground so there's like some stairs that get up to this mounting position for the ferris wheel and then after the After the ferris wheel rotates a little bit His little bucket seat moves up here to point one P one in which case then the next group can get on to that seat as well So he's sitting there and our rider is thinking how high above the ground Would I be right now? Assuming that the angle that the the central angle of this circle is going to be 45 degrees How high is the rider going to be so what is this distance h right here? So I want to present to you a different diagram that's going to clean up this model a little bit here So what we hear about here is the circle the buckets the ramp the stairs They know that thing really matters what really matters is the following observations here So we know that the circle is 14 feet above the ground that's going to affect this distance h We also know that the circle has a radius of 125 feet So these two distinct radii in the prom are both going to be 125 feet And we know that the central angle from where our riders started to where they're currently parked on the ferris wheel That was 45 degrees. So these are the measurements we have in play here. How is that going to help us find h? So what we have to do is we have to form a right triangle We don't want to use this triangle right here because it's not a right triangle Nor do we want to use the sector of the circle associated to points P 9 and P 1 because that's not a triangle either It's got some curvature to it So what we want to do is we actually want to consider take the horizontal line Why we do in horizontal line because it's parallel to the ground so take the horizontal line that goes through P1 and it'll intersect the original radius at this right angle right here Okay, now this triangle in play is a right triangle and I'm going to pull the picture over here So we have this right triangle the angle measure here is 45 degrees We know that the radius of the circle is 125 feet So that'll be the hypotenuse of this right triangle and let's look at this unspecified number x All right, we'll get we'll get to x why we care about x just a moment So we want to figure out what's this distance here this follows from a very basic basic cosine ratio So notice that we get x over 125. This is going to equal cosine of 45 degrees for which if we solve for x we're going to get that x equals 125 times cosine of 1445 degrees like so so that figures out this distance x well Why does that matter well the reason we care about x is because if we go if we calculate the distance from the center of the Ferris wheel to the ground we know that distance because this distance is 14 and this distance is 125 so the total distance above the ground to get to the center of the ferris wheel notice that's going to be 14 plus 125 okay, but notice that if we take the distance from the center of the circle to The bottom of the ground there that's going to be x plus h. So this number right here is equal to x plus h So the number h is what we want to know x is something we found using trigonometry So solving for h h is going to equal 14 plus 125 minus x for which we now see that x is equal to Just plug plugging it in here We're going to end up with negative 25 times cosine of 45 degrees So at this moment we can then start to simplify the calculation again This is something you put into your calculator. Of course 14 plus 125. It's going to be 139 Put into your calculator cosine of 45 degrees. Make sure it's in degrees. That'll give you approximately 0.0 707 times that by 125 and subtract you're going to get your estimate to be approximately 50 point Six feet in the air. That's the that's the passenger's current location But that's based upon right this one right but as he goes to different places on our ferris wheel Let me move this picture back up If as the individual as a writer keeps on going around and around and around and around the height will change based upon where Is in the circle and so what happens if we do a different location, right? What if the central angle this time is this one right here? Can we think of the height of the passenger with respect to this central angle? And that's that's of course going to be the case by mimicking our strategy from before we see that h Is going to equal 14 plus 125 minus 125 125 times cosine of theta for which you can factor out the 125 and you're going to get the h equals 14 plus 125 Times one minus cosine of theta And so that gives a formula of the height of the passenger given that central angle but As 125 which is the radius of the circle if we know the radius And if we know this distance right here sort of like the the bottom height of the of the circle right there We'll call it y we see that the height of the passenger is going to be well How high above the ground is the ferris wheel? Plus the radius of the ferris wheel times one minus cosine times theta of the central angle This formula would tell you the height on the ferris wheel for the rider in any moment given that information So basically if you're trying to measure Uh, what's the height above the ground if you're on a circle and you know the central angle Then you can find it by this formula. It just follows from basic So katoha right triangle trigonometry. So that's going to then conclude our Lecture here lecture number five about applications of right triangle trigonometry. Thanks for watching everyone I hope you learned something if you did please give this in all these other videos a like If you want to see more videos like this in the future, of course subscribe to the channel and by all means If you have any questions, please post those in the comments below I'll be glad to answer them when I have the opportunity. Bye everyone