# 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 the rider is 16 m above ground level?

Great problem. It's a new-to-me kind of related rates.

Here is a sketch.

I'm sorry I can't label it clearly, but I'll talk through it.

The wheel is tangent to the ground and I've drawn a center line at

The rider is at the blue dot.

The radius (from

The height above the line

The central angle from the positive horizontal

The height above ground is

That is the same as finding

So we have:

Variables

Rates of change

Equation relating the variables:

Solve the problem:

When

So at that point

Notes for students

I've taught university and college calculus for years, but I had never seen this problem. So I had to approach it fresh.

You may find help and encouragement from these notes.

I started with a sketch of the wheel with its rider and the angle of elevation from the bottom of the wheel.

That led to some rather ugly calculations They could be done, but they were challenging.

In thinking about the rate of change of the angle I realized that I'd be better off staring my picture when the rider is 10 m above the ground.

Then the height became

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To find how fast the rider is rising, we can use the formula for the rate of change of height with respect to time, which is given by:

[ \text{Rate of change of height} = \frac{{\text{Radius} \times \text{Angular speed}}}{{2\pi}} ]

Given: Radius (r) = 10 m Angular speed (ω) = (\frac{1}{2}) revolutions per minute (since one revolution every 2 minutes)

Now, substitute the values into the formula:

[ \text{Rate of change of height} = \frac{{10 \times \frac{1}{2}}}{{2\pi}} ]

[ \text{Rate of change of height} = \frac{5}{\pi} \approx 1.59 \text{ meters per minute} ]

So, the rider is rising at approximately 1.59 meters per minute when the rider is 16 meters above ground level.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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