Assumptions in Word Problems (Calculus)

Question

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?

Answer 1

A man walks into a restaurant, orders food, eats, then leaves.

Did the man have to cook his food? Did he pay? Did he sit down? Did he eat the food he ordered or just eat something else entirely.

Your problem is known in Psychology as the interpretation and application of one's schema. http://en.wikipedia.org/wiki/Schema_(psychology)

Because people are different, they often "fill in the gaps" using different schema than the problem designer had intended. I have experience with this as I used to teach.

If you are ever in doubt. Ask the instructor, or try to imagine the simplest solution possible (no leak or explosions or any other outside problem).

Answer 2

Some word problems have unstated assumptions that are actually problematic. Here's one of my favorite examples:

Alice can paint a wall in one hour and Bob can paint it in two. How long will they take to paint it together?

The intended interpretation is that they paint the wall at a constant rate, and that they work together in such a way that you can just add together their painting speeds (like if they're starting at opposite ends of the wall and coming together). This is a somewhat artificial interpretation, and in my opinion not entirely obvious.

I think your balloon problem is a terrible example, though. Of course the balloon isn't leaking, that information would have been included in the problem. That's not an ambiguity of the problem statement, that's you making up possible things it could be leaving out. I think (and I'm only speculating) that you might have gotten burned on a few genuinely ambiguous word problems, and it's made you paranoid about all of them. Take a step back, breathe, and remember your common sense.

Answer 3

Too big for a comment:

You can assume that the gas leaks but the leakage is negligible so that the numbers in the problem hold and you can solve it, if it is not negligible then the problem changes.

What you are doing is solving the problem with pen and paper, i.e. theoretically, so we discard unwanted factors (such as temperature, higher temp increases the volume of the gas) but we did not consider it in our problem because for the problem under consideration it does not matter.

In real life there can be hundreds of external parameters and also practical limitations like the gas tank is out of gas, will you be able to solve the problem under consideration if you assume this situation?

Answer 4

Everything has a context.

If that question was asked in the context of a class about balloon physics, you could bet that you would have to use concepts taught therein, such as rubber elasticity, gas densities, shape properties, etc.

Because this is a calculus class, if you knew the material well, you would easily deduce that the problem most likely is asking you to apply a straightforward differential because, if there was a leak, then that would most likely fall into the differential equations realm. Also, you would have most likely run into a similar homework problem.

Answer 5

Interestingly,

For anyone else who stumbles upon this question, the answer is rather simply.

Take a look at,

The Cooperative Principle of Communication

The Maxim of QUALITY states one should not mention what is false.

Which is the answer to the question.