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# A Ferris wheel with a radius of $10 m$ is rotating at a rate of one revolution every 2 minutes. How fast is a rider rising when his seat is $16 m$ above the ground level?

## 8$\pi$ meters per minute

Derivatives

Differentiation

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### Video Transcript

So we want to know how fast the writer is rising when the seat is 16 m above the ground level. So if we want a picture are fair squeal is just being a circle. We know that the radius is 10 m and then we also you can see that since this right here would be 10 m if say, there's 16 ft off the ground At that point, we have a right triangle right here, or this is Theda and we want to find, um this. So with all this in mind, we see that the height of the writer any time is going to be X plus the 10 m because this is 10 m from the ground. So with that in mind, we see that the sign of data is equal to X over 10. This is our X value. This is the 10 that we're talking about eso. Since the sign of data is X over 10, we differentiate both sides and see that the co sign of Fada Since the derivative assigned data is coasting, data be at cosine theta d theta DT is equal to 1/10 dx DT then, um, we can rewrite co sign to keep it in terms of sign. So that's the same thing as the square root of one minus sine squared data de theater. DT is equal to once again 1/10 of DX DT. So when the height of the writer is 16 m, we know that X is equal to six. So in that case, we would have since X is equal to six. Sign of data is equal to 6/10. So r theta value would equal the inverse sign of, um, 3/5 or 6/10. Um, and what we'd end up getting as a result is that sign of Fada equaling 3/5 would give us, um since we know that the fairy were the Fairfield makes one revolution for two minutes. Do you think the DT equals two pi radiance over two minutes? So that means that there is pie radiance per minute. So that's the rate deflated ET so substituting these values and now what we end up getting is a square root of one minus nine. 25th Now, because it's sign as squared, which we already know, Sign of data is 3/5. Then we're gonna have times pi r rate is gonna be equal to 1/10 dx DT So with that, we just do the math and simplify and we end up getting that DX DT is equal. Thio ate pie. So what that tells us is that, um, the writer is traveling ate pie meters permanent.

California Baptist University

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