6 of 6 Items .... Source: Kate Nowak

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The last of the three related-rate geogebra problems from Kate Nowak. It's the related rate problem from calculus: the conical tank being filled with water.

Adjust the slider and... wait, what is changing and how?

For every click of the slider:
Is the depth increasing at a constant rate?
Is the radius increasing at a constant rate?
Is the volume increasing at a constant rate?
How can you tell?

• Where or how, in the RealWorld^tm, could we see the constant increase in volume?


• Where or how, in the RealWorld^tm, could we see the constant increase in radius, or depth?

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We've all seen this problem, but many of our students haven't. It's the related rate problem from calculus: the balloon being filled with air.

There are two questions being demonstrated here.
(1) "If the volume increases at a constant rate, what is happening to the radius?" and
(2) "If the radius increases at a constant rate, what is happening to the volume?"

The first question is to figure out which situation is modeled in red and which in blue.
Then we can ask:

• Does the radius increase at a constant speed in both models? How can you tell?


• Does the volume increase at a constant speed in both models? How can you tell?


• Where or how, in the RealWorld^tm, could we see the constant increase in volume?


• Where or how, in the RealWorld^tm, could we see the constant increase in radius?

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We've all seen this problem, but many of our students haven't.

It's the related rate problem from calculus: the ladder sliding down the wall.

The "official" question?

How fast is the ladder's top sliding down the wall if the bottom is being pulled out at a rate of 1 ft/sec?

We can ask a few questions of kids at any level, though, based on the given that the bottom of the ladder is being pulled to the right at 1 foot per sec.

• Does the top drop at a constant speed?


• Does the top drop a distance equal to the horizontal movement?


• When is the speed of the top greater than 1, less than 1, and equal to 1?


• If this is a 25 foot ladder, with the bottom 7 feet out from the base of the wall, and the top drops 4 feet ... how far out does the bottom of the ladder have to go?

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\( x^2 = 81 \) has two solutions, -9 and 9.

But does \( \sqrt{81} \) have one solution or two?

Is it correct to say that \( \sqrt{81} \) = +9 and -9?

Or should we be saying that \( \sqrt{81} \) is an expression and that \( 9\), \(\frac{18}{2} \), \(27^{2/3} \), and \(1+6+2\), are equivalent expressions?

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Are these the same or not the same? Why? (What makes them different if you think they're different?)

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Can you show if this is true?

\(2^6 * 2^6 = 2^{11} + 2^{11}\)

Can we generate more like this one ... simple and elegant?
How about one with 3 as a base?