Trying to come up with a calculus problem

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Cal San
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Trying to come up with a calculus problem

Postby Cal San » Wed Feb 18, 2009 5:45 pm UTC

Okay, here's the basic idea.

You have a cylindrical glass of water. The glass has a radius of 2 cm, and is filled to the height of 15 cm. You are emptying it by a straw, dipping it into the bottom of the glass, and putting your finger over it, moving it out of the glass, and letting your finger off the top. This straw has a radius of .3 cm. You take the water out at one straw every 5 seconds. How long will it take for the glass to be down to a volume of 20*pi?

-any idea of :
a)How to ACTUALLY set up this equation, if any?
b)How to improve the question?
c)If I have any idea what I'm talking about?
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NathanielJ
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Re: Trying to come up with a calculus problem

Postby NathanielJ » Wed Feb 18, 2009 6:02 pm UTC

Cal San wrote:Okay, here's the basic idea.

You have a cylindrical glass of water. The glass has a radius of 2 cm, and is filled to the height of 15 cm. You are emptying it by a straw, dipping it into the bottom of the glass, and putting your finger over it, moving it out of the glass, and letting your finger off the top. This straw has a radius of .3 cm. You take the water out at one straw every 5 seconds. How long will it take for the glass to be down to a volume of 20*pi?

-any idea of :
a)How to ACTUALLY set up this equation, if any?
b)How to improve the question?
c)If I have any idea what I'm talking about?


As far as I can tell, there's no calculus needed here at all (well, besides perhaps the implicit calculus in the fact that you're using pi). It's a discrete problem, not a continuous one.

Hint 1:
Spoiler:
The thing to note is that each time you take out a strawfull of water, you are removing a fixed proportion (say p) of the total remaining water. That p is easy enough to calculate (p = 0.0225).


Edit: removed second hint due to potential homeworkness. Is this at all for homework?
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Busdriver
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Re: Trying to come up with a calculus problem

Postby Busdriver » Thu Feb 19, 2009 3:48 am UTC

Here are a few ways you might improve the problem. First, if you want it to be a calculus problem you should probably drain it out of the straw continuously. Second, if you want it to be a bit more challenging, maybe change the shape of the glass. You could try a cone or a sphere or something, even though they wouldn't make much sense as glasses. If you want it to still drain at a variable rate, you could maybe have it drain out of a hole at the bottom at a rate based on the pressure from above. Though I'm not really sure how that would work.

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MartianInvader
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Re: Trying to come up with a calculus problem

Postby MartianInvader » Thu Feb 19, 2009 4:11 am UTC

A standard sort of problem you might see in second-semester calculus: "How much work is done draining the water out through the straw if the straw sticks 5cm above the top of the glass?"
Let's have a fervent argument, mostly over semantics, where we all claim the burden of proof is on the other side!

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Cal San
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Re: Trying to come up with a calculus problem

Postby Cal San » Thu Feb 19, 2009 3:54 pm UTC

Thanks for all the help so far.
To answer your question, NathanielJ, no this is not for homework. Right now in my calc BC class I'm doing more of reduction formulas and integrals of special powers of trig functions.

As for turning it into a calculus problem, I wanted to try to make it more than just a dv/dt problem. Maybe I could say that the straw is draining it at the rate of kv cm^3 /s. But then it would just be related rates, and the width of the straw would be irrelevant.

By the way...
MartianInvader wrote:A standard sort of problem you might see in second-semester calculus: "How much work is done draining the water out through the straw if the straw sticks 5cm above the top of the glass?"

-wouldn't that be more physics? After all, work is a function of force and distance, is it not? And force is a physics thing, right? I guess it might use calculus to figure it out, but still.

Again, I just want to come up with a calculus problem. Not for a class, not for anyone else, just for me. Does that make me strange?
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Talith
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Re: Trying to come up with a calculus problem

Postby Talith » Fri Feb 20, 2009 2:15 am UTC

One of my favourite calculus questions (although from your last reply you may think it's more physical) Is to find a differential equation, and thus a solution of, the shape of a slack, hanging chain/rope (held at two points at an equal height) when under gravity.

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Cal San
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Re: Trying to come up with a calculus problem

Postby Cal San » Fri Feb 20, 2009 2:10 pm UTC

Yeah I think i do find it more physical. But it does sound like a good problem.
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btilly
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Re: Trying to come up with a calculus problem

Postby btilly » Fri Feb 20, 2009 3:03 pm UTC

I've always been fond of asking for the dimensions of the cylinder that encloses a maximum volume with a fixed area. Then go to the store, get a real soup can, compare its dimensions, study its exact shape, and figure out the various reasons why might be better than the apparently optimal shape you calculated. (Trust me, the design of your standard soup can has been fairly carefully optimized.)
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