Assumptions in Mathematical (Calculus) word problems

[Amad27]

Hi,

 

Hello,

I just had a small question about assumptions in mathematical word problems. Suppose you are given a calculus problem (related-rates),

"A spherical balloon is inflated with gas at the
rate of 800 cubic centimeters per minute. How fast is the radius of the balloon increasing at the instant the radius is (a) 30 centimeters and (b) 60 centimeters?"(Larson Calculus P 153)

This is a quoted problem from Larson's Calculus 10th Edition.

My question is here:

Why do you assume for example that there is no hole from, which air LEAVES?
Basically, in general, why do you make unstated assumptions for example, there is no air leaving, or the balloon doesnt explode before the radius is (a) 30 cm etc..?

Here is what others say,

Others say it is so you could solve the problem, what do you think?

 

Thanks a lot =)

[imatfaal]

Otherwise you would never finish writing questions - assume there is no alien invasion, assume that it is 3d space, assume that THEY do not shoot anyone who does calculus, assume that pi is the usual one and not 3, assume... assume... etc.

 

Science tends to be about making simplifying assumptions and creating models that work under these assumptions - you then test to see if these models work under experimental conditions.

 

To answer your question - you make those assumptions necessary to answer the question as asked. If the question requires a significant assumption then you should note this in your answer - ie if the word spherical had been omitted I would mention that I was assuming that the balloon stayed spherical as inflated. If it was a physics question I might also make a few comments about the changing pressure (or not) within the balloon.

[Unity+]

Why do you assume for example that there is no hole from, which air LEAVES?

Basically, in general, why do you make unstated assumptions for example, there is no air leaving, or the balloon doesnt explode before the radius is (a) 30 cm etc..?

It's less of an assumption and more of a conclusion based on deductive reasoning. It's like asking "Well, we may know what is in the set, but then what is not in the set?" You are asking a question that could have an infinite amount of possibilities. Therefore, the logic is not reasonable to follow.

[Amad27]

Otherwise you would never finish writing questions - assume there is no alien invasion, assume that it is 3d space, assume that THEY do not shoot anyone who does calculus, assume that pi is the usual one and not 3, assume... assume... etc.

 

Science tends to be about making simplifying assumptions and creating models that work under these assumptions - you then test to see if these models work under experimental conditions.

 

To answer your question - you make those assumptions necessary to answer the question as asked. If the question requires a significant assumption then you should note this in your answer - ie if the word spherical had been omitted I would mention that I was assuming that the balloon stayed spherical as inflated. If it was a physics question I might also make a few comments about the changing pressure (or not) within the balloon.

Thanks,

 

The basic idea is making assumptions is required otherwise we will never finish anything?

It's less of an assumption and more of a conclusion based on deductive reasoning. It's like asking "Well, we may know what is in the set, but then what is not in the set?" You are asking a question that could have an infinite amount of possibilities. Therefore, the logic is not reasonable to follow.

Right,

 

So you make that assumption based on thinking that if you didn't there could be infinite other things to think about? If you had to state it in ONE sentence, what would you say? THANKS

[Unity+]

Right,

 

So you make that assumption based on thinking that if you didn't there could be infinite other things to think about? If you had to state it in ONE sentence, what would you say? THANKS

Assumption: A thing that is accepted as true or as certain to happen, without proof

 

I would think that my statement is not an assumption, but a generally accepted statement that is proof of itself, almost like an axiom.

[Amad27]

Assumption: A thing that is accepted as true or as certain to happen, without proof

 

I would think that my statement is not an assumption, but a generally accepted statement that is proof of itself, almost like an axiom.

Hello,

 

That is excellent what you stated. So thus, the BOTTOMLINE is that,

 

You make certain assumptions based on no reason. Accepting it axiomatically?

 

Thank you so much for your help, I really appreciate it, really appreciate it.

-Thanks, Aakarsh

[Unity+]

Hello,

 

That is excellent what you stated. So thus, the BOTTOMLINE is that,

 

You make certain assumptions based on no reason. Accepting it axiomatically?

 

Thank you so much for your help, I really appreciate it, really appreciate it.

-Thanks, Aakarsh

No, you are putting words into my mouth. I never said it was an assumption.

[Amad27]

No, you are putting words into my mouth. I never said it was an assumption.

 

Hi,

 

I think I misunderstood then.

 

Which statement were you talking about that is generally accepted then?

 

Thank you so much, I really, really appreciate it.

[pzkpfw]

"based on no reason" seems an unfair addition too.

 

Assumptions for a given case may be without proof (for that case), but may be based on other factors - like past experiences.

 

For example, I've never blown up a balloon with any hole large enough to notice at the time. So given the problem in post #1 I probably wouldn't think about air escaping as the balloon was inflated. Most people, I would assume, would consider that a reasonable assumption (no hole, no air escape).

 

It's possibly true that a cleverer person than me would have thought of that possibility, but on the other hand the experience of being given questions in a text book or exam also apply for assumptions - it'd be clear to most people what the question is asking, and that the answer is supposed to come from the parameters given - that it isn't required to make guesses about other factors, that it can be assumed there are no other factors.

 

If you were answering that question in an exam, they didn't tell you about a hole in the balloon - so why would you invent one? And what rate of air escape would you assign it? That there is no hole seems a valid assumption, in the context of the question.

 

 

I would ask, Amad27, why do you ask this question about assumptions?

Edited August 19, 2014 by pzkpfw

[Amad27]

"based on no reason" seems an unfair addition too.

 

Assumptions for a given case may be without proof (for that case), but may be based on other factors - like past experiences.

 

For example, I've never blown up a balloon with any hole large enough to notice at the time. So given the problem in post #1 I probably wouldn't think about air escaping as the balloon was inflated. Most people, I would assume, would consider that a reasonable assumption (no hole, no air escape).

 

It's possibly true that a cleverer person than me would have thought of that possibility, but on the other hand the experience of being given questions in a text book or exam also apply for assumptions - it'd be clear to most people what the question is asking, and that the answer is supposed to come from the parameters given - that it isn't required to make guesses about other factors, that it can be assumed there are no other factors.

 

If you were answering that question in an exam, they didn't tell you about a hole in the balloon - so why would you invent one? And what rate of air escape would you assign it? That there is no hole seems a valid assumption, in the context of the question.

 

 

I would ask, Amad27, why do you ask this question about assumptions?

 

I am sorry, I had been trying to answer this question myself, I just thought of this question, and was stumbled; I had never thought of it before.

 

But would you agree with me if I said,

 

Someone else told me, when interpreting a question (word problem) you choose the SIMPLEST INTERPRETATION. Which is without a hole to be exact.

 

Bottomline: Is this true?

[studiot]

 

Someone else told me, when interpreting a question (word problem) you choose the SIMPLEST INTERPRETATION. Which is without a hole to be exact.

 

Bottomline: Is this true?

 

 

No.

 

pzkpfw has already told you the simple answer.

 

A question of the sort you posed is meant to be answered using only the information given.

 

If a piece or pieces of information are missing so any solution is impossible, the question is at fault.

 

A word of advice.

 

In the past students could expect to need to use every piece of information to solve all the question.

So if you had a piece of information 'left over' unused then you could be sure you have made a mistake and missed something.

 

I have seen some modern questions that contain unecessary information as a deliberate distraction.

Edited August 19, 2014 by studiot

[pzkpfw]

This all reminds me of a "story" told to me by my physics teacher, back in '86. I gather it's a common story, but don't know the details (I presume it's a myth, for illustration of a point, not based on an actual event).

 

I won't try to tell that story in any entertaining (or accurate) way ...

 

It involves a physics exam, and one question is along the lines of "you have a barometer; how do you measure the height of a building?". Most of the class provides an answer based on the difference in air pressure at the bottom and top of the building. Not an unreasonable assumption based on the equipment given.

 

But this one kid gives an answer along the lines of dropping the barometer off the top of the building, and measuring the time taken. The teacher doesn't accept this as it's not the expected answer, so makes the student re-take the test. There's a whole lot more methods the kid comes up with, like unravelling his jersey and using the barometer as a weight to dangle the wool from the top of the building. I don't remember them all.

[Amad27]

This all reminds me of a "story" told to me by my physics teacher, back in '86. I gather it's a common story, but don't know the details (I presume it's a myth, for illustration of a point, not based on an actual event).

 

I won't try to tell that story in any entertaining (or accurate) way ...

 

It involves a physics exam, and one question is along the lines of "you have a barometer; how do you measure the height of a building?". Most of the class provides an answer based on the difference in air pressure at the bottom and top of the building. Not an unreasonable assumption based on the equipment given.

 

But this one kid gives an answer along the lines of dropping the barometer off the top of the building, and measuring the time taken. The teacher doesn't accept this as it's not the expected answer, so makes the student re-take the test. There's a whole lot more methods the kid comes up with, like unravelling his jersey and using the barometer as a weight to dangle the wool from the top of the building. I don't remember them all.

 

Actually,

 

What you say here is critical. Check out,

 

http://www.sas.upenn.edu/~haroldfs/dravling/grice.htmlGrice's Maxims

 

The Grice Maxim of Quality states, only say what you believe is TRUE, DONT say what you believe is false.

 

Communication is dependent on the Grice Maxim, which is why this was developed because this is ordinary human communication.

In word problem communication the author DOESNT state what is false, but DOES state what is TRUE.

 

Very interesting, what do you think?

[Amad27]

"based on no reason" seems an unfair addition too.

 

Assumptions for a given case may be without proof (for that case), but may be based on other factors - like past experiences.

 

For example, I've never blown up a balloon with any hole large enough to notice at the time. So given the problem in post #1 I probably wouldn't think about air escaping as the balloon was inflated. Most people, I would assume, would consider that a reasonable assumption (no hole, no air escape).

 

It's possibly true that a cleverer person than me would have thought of that possibility, but on the other hand the experience of being given questions in a text book or exam also apply for assumptions - it'd be clear to most people what the question is asking, and that the answer is supposed to come from the parameters given - that it isn't required to make guesses about other factors, that it can be assumed there are no other factors.

 

If you were answering that question in an exam, they didn't tell you about a hole in the balloon - so why would you invent one? And what rate of air escape would you assign it? That there is no hole seems a valid assumption, in the context of the question.

 

 

I would ask, Amad27, why do you ask this question about assumptions?

 

 

So, I have a last question on what you said. Is the bottomline that,

 

You make certain assumptions because you are assuming the problem **IS** SOLVABLE?

[ajb]

You make certain assumptions because you are assuming the problem **IS** SOLVABLE?

In a school or university test or exam you would usually assume that the question is solvable in the sense I think you mean. It is also possible that the answer is "there is no solution", i.e. one cannot get numerical answers to the posed question.

 

Wider than just examination type questions, in applied mathematics one will need to make some initial simplifying assumptions. Without these assumptions you may not be able to mathematically model the systems you are interested in. From there you would like to solve the equations you have constructed. To do this you may need to make further assumptions so that you can get a closed form or some other form you can handle as a solution to your simplified equations. Or you could turn to numerical methods...

 

Anyway generically you like to make simplifying assumptions, but as few as possible to get a mathematical system you can deal with.

[Amad27]

In a school or university test or exam you would usually assume that the question is solvable in the sense I think you mean. It is also possible that the answer is "there is no solution", i.e. one cannot get numerical answers to the posed question.

 

Wider than just examination type questions, in applied mathematics one will need to make some initial simplifying assumptions. Without these assumptions you may not be able to mathematically model the systems you are interested in. From there you would like to solve the equations you have constructed. To do this you may need to make further assumptions so that you can get a closed form or some other form you can handle as a solution to your simplified equations. Or you could turn to numerical methods...

 

Anyway generically you like to make simplifying assumptions, but as few as possible to get a mathematical system you can deal with.

 

Actually,

 

The question of whether a problem is solvable (numeric answer) does not depend on it being a university or school problem, but it lies in this,

"Find...."

 

In the related-rates for example; it means that you can find the rate so it is solvable.

 

If a problem is not solvable, then you CAN make assumptions (of holes etc..)

 

@ajb, can you help me with something?

 

When you are first ever taught word problems, they dont teach why you don't assume there is a hole (from which air leaves) etc...

Then what is the actual reason?

 

Almost everyone I have talked to has said it is so that the problem is solvable; meaning you CAN find the answer (numeric etc).

Help?

[ajb]

In the related-rates for example; it means that you can find the rate so it is solvable.

As it is taken from a textbook I am sure you can find the rate given the information given in the question.

 

If a problem is not solvable, then you CAN make assumptions (of holes etc..)

Just practically, you may need to make further assumptions. However for the standard textbook these assumptions will either be already made for you (eg. spherical balloon, constant rate of gas in, no gas leaving etc) or they will be rather standard, but you should still state them clearly.

 

When you are first ever taught word problems, they dont teach why you don't assume there is a hole (from which air leaves) etc...

Word problems students always find difficult. Often a straight forward calculation is easy for them, but a word question can stump good students. So you are not alone in this, far from it. I too always hated "wordy questions" in mathematics classes.

 

 

Then what is the actual reason?

The reason is just to not over complicate the question. You could keep adding further things, like the rate of gas in is not constant in time, some gas escapes and so on. Maybe you can still solve this problem explicitly with these further things, maybe you can't or maybe it would just be too long and complicated for a good question to ask students.

 

Almost everyone I have talked to has said it is so that the problem is solvable; meaning you CAN find the answer (numeric etc).

Essentially this is the case. That is not to say that a more complicated set of assumptions necessarily has no numerical solution.

 

Why are you asking about such questions? You are a student?

Edited August 21, 2014 by ajb

[Amad27]

Hello,

 

I am a student (15), and I have OCD, so I try to find satisfying answers.

 

Mathematics matters the MOST to me. It's my absolute best and favorite subject.

 

It is because I like math, I try to get the clearest vision I can for it. I try to excel more and more at it =)

 

That is why I asked this question. I was confused about this for quite a while.

 

I am still stumped. For some reason, my brain just can't accept that this is because you shouldn't complicate a problem.

 

The only issue I have with this is that, how would one find out if the correct answer is the simplest or the complicated way. That is why I cant work this out properly.

 

Like on a test, how would you know if the teacher is looking for the simplest or the complicated way?

[ajb]

The only issue I have with this is that, how would one find out if the correct answer is the simplest or the complicated way. That is why I cant work this out properly.

Quite often there is more than one way to get at the right answer. It may not be clear which answer really is the simplest.

 

For example, you may find the solution to some problem obvious once you know some higher mathematics. But that is not really "simpler" than using lower brow mathematics as one needs a lot in place already to write doen the "obvious" solution.

 

Like on a test, how would you know if the teacher is looking for the simplest or the complicated way?

Generally I would say go for the simplest, but that is not a clear statement as people do think differently and that can be seen in their solutions to questions. As a teacher, any reasonable method that works should be marked as correct. At your level I expect there not to be too many different way to tackle a question. Also, your teacher would have shown you one or two methods, but students can suprise you with solutions you are not expecting. This can make allocation of marks not so clear if the student makes mistakes but is on the right lines.

[Amad27]

Quite often there is more than one way to get at the right answer. It may not be clear which answer really is the simplest.

 

For example, you may find the solution to some problem obvious once you know some higher mathematics. But that is not really "simpler" than using lower brow mathematics as one needs a lot in place already to write doen the "obvious" solution.

 

 

Generally I would say go for the simplest, but that is not a clear statement as people do think differently and that can be seen in their solutions to questions. As a teacher, any reasonable method that works should be marked as correct. At your level I expect there not to be too many different way to tackle a question. Also, your teacher would have shown you one or two methods, but students can suprise you with solutions you are not expecting. This can make allocation of marks not so clear if the student makes mistakes but is on the right lines.

I think I am getting there.

 

So there can definitely been 2+ ways to do the word problem (which may or may not) always give the correct answer.

 

So it is up to you to choose, which way you want to go right?

 

Thank you ajb (once again)!

Edited August 21, 2014 by Amad27

[ajb]

So there can definitely been 2+ ways to do the word problem (which may or may not) always give the correct answer.

There may be many ways to solve any mathematical problem, including those of applied mathematics, physics and so on.

 

So it is up to you to choose, which way you want to go right?

Yes, unless the question very specifically tell you to us a given method.

[Amad27]

There may be many ways to solve any mathematical problem, including those of applied mathematics, physics and so on.

 

 

Yes, unless the question very specifically tell you to us a given method.

 

Hello,

 

I just had a question for you, since you seem like the one who wants to help me =)

 

I posted this same thing on another forum.

 

They said you make certain assumptions because of how you learned to communicate.

 

Can you look at the Grice Maxims I posted earlier? And share your thoughts on it.

 

Thanks (and thanks for your commitment)

[ajb]

Grice's Maxims seem to be good rules to follow when constructing a mathematics question for students. My quibble with "you make certain assumptions because of how you learned to communicate" is that these assumptions are needed in order for you to first formulate the questions and secondly to solve the problem. This is independent of the communication of mathematics, which in this case is related to explicitly stating the assumptions.

 

I really do think you are thinking too hard about this. I don't see that there is anything mathematically deep here nor deeply associated with how we communicate.

[Amad27]

Grice's Maxims seem to be good rules to follow when constructing a mathematics question for students. My quibble with "you make certain assumptions because of how you learned to communicate" is that these assumptions are needed in order for you to first formulate the questions and secondly to solve the problem. This is independent of the communication of mathematics, which in this case is related to explicitly stating the assumptions.

 

I really do think you are thinking too hard about this. I don't see that there is anything mathematically deep here nor deeply associated with how we communicate.

 

Hello,

 

Thanks for the reply @ajb.

 

My only one regard with this is that, don't you think interpreting a problem is in fact, communication, which does in fact required communication skills?

 

Which in then does use the Maxims. That is all I thought of, that is important.

 

Thanks for helping =)

[ajb]

My only one regard with this is that, don't you think interpreting a problem is in fact, communication, which does in fact required communication skills?

 

Which in then does use the Maxims. That is all I thought of, that is important.

You are absolutly right, but this is now a question of communication of the assumptions and not a question of why we need to make assumptions in the first place. Having well-posed mathematical questions is not the same as presenting them in a clear way!