Video: Constructing and Solving Trigonometric Equations Modeling Real-Life Situations Involving Ferris Wheels

A Ferris wheel is 20 m in diameter. A ride takes 6 minutes and consists of one complete revolution, starting and finishing at the lowest point. When riders board the Ferris wheel, their seats are 2 m above the ground. How much of a ride is spent more than 13 m above ground?

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Video Transcript

A Ferris wheel is 20 meters in diameter. A ride takes six minutes and consists of one complete revolution, starting and finishing at the lowest point. When riders board the Ferris wheel, their seats are two meters above the ground. How much of a ride is spent more than 13 meters above ground?

For problems like these, it’s often helpful to sketch what’s happening here. We have a circular Ferris wheel. The ride starts and ends at the lowest point. This lowest point is two meters from the ground. And the diameter, the distance from one end of the circle to the other through the middle, is 20 meters. This means the highest point of the Ferris wheel is 22 meters. We can also break up the distance across into 10 meters from the center to one side and 10 meters from the center to the other side. This tells us that the center of the Ferris wheel is a height of 12 meters. And our question is how much of the ride is spent more than 13 meters above ground. We can sketch a line that’s 13 meters high, which is just above the center of the Ferris wheel. The arc of the circle, here, is the time that people spend above 13 meters.

There are a few other things we need to remember. In a circle, one complete revolution equals 360 degrees. And so we can set up this ratio for our circle. In six minutes, you go 360 degrees. We can label the point where people hit 13 meters as 𝐴 and the point where they hit 13 meters going down as point 𝐡. We can call the space where the Ferris wheel is above 13 meters arc 𝐴𝐡. And we want to know how many minutes are spent going around the measure of arc 𝐴𝐡. And that means we need to know how many degrees is arc 𝐴𝐡.

To do that, we could use a central angle. A central angle is an angle formed by two radii, with the vertex at the center of the circle. And so we connect point 𝐴 to the center of the circle. And we connect point 𝐡 to the center of the circle. If we call the center 𝑂, then we can say we have a central angle at angle 𝐴𝑂𝐡. Why is this important? The central angle is equal to the intercepted arc. And so we say that the measure of angle 𝐴𝑂𝐡 equals the measure of arc 𝐴𝐡. If we can find the measure of angle πœƒ, the angle between 𝐴𝑂𝐡, we’ll know how many degrees are spent above 13 meters.

At this point in our diagram, we don’t need this 10-meter radius. We know that the distance from the center to the 13-meter line is one meter. And then, we notice that, here, we have a right angle. We’ve already said that the distance between the center and the 13-meter line is one. And we know that the radius is 10 meters. If we can find this angle, we’ll be able to double it to find angle 𝐴𝑂𝐡. So this is what we have, a right triangle. And we know the adjacent side length and the hypotenuse side length.

Since we’re dealing with a right-angled triangle and we have the adjacent and the hypotenuse side lengths, we recognize this as the cosine relationship. If we call that angle, angle 𝑏, we can say that cos of 𝑏 is equal to one over 10. To find out the measure of angle 𝑏, we take the cos inverse of cos 𝑏 and the cos inverse of one 10th. The cos inverse of cos 𝑏 is just 𝑏. And the cos inverse of one 10th equals 84.2608 continuing degrees. And our central angle, angle 𝐴𝑂𝐡, is two times this amount. This is because the right triangle created on the left side is the same size as the triangle created on the right side.

When we multiply the angle we found by two, we get 168.521659 continuing degrees. And so we can say that the central angle is 168.521 continuing degrees. And remember, the central angle in our arc 𝐴𝐡 have the same measure. And so we can solve our proportion. If it takes six minutes to go 360 degrees, it will take π‘₯-minutes to go 168.521 continuing degrees. We can use the variable π‘₯ to represent the minutes we’re trying to solve for. And so we can multiply six times 168.521 continuing degrees.

Notice that we haven’t rounded this value yet. We’re using this unrounded form because we want to wait until the final step to round to produce the most accurate answer. You can do this easily by storing the value in your calculator. The function usually looks something like this. Multiply six by the previous answer. This gives me 1011.129 continuing. Now, we need to solve for π‘₯ by dividing both sides by 360. When we do that division, we get 2.8086 continuing equals π‘₯.

Remember that π‘₯ is a measure of minutes. And we don’t usually use decimal values. We wouldn’t usually say 2.8 something minutes. We would say two minutes and some seconds. To find the number of seconds, we take the decimal value 0.8086 continuing minutes. And we multiply it by 60 because there are 60 seconds in one minute. When we do that, we get 48.521 continuing. So we have 48 seconds and this decimal value as a partial second. It seems reasonable now that we would round up to the nearest second. We have about 48 and a half seconds just over that. And so we can round to 49 seconds.

On this ride, you would spend two minutes and 49 seconds above 13 meters.

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