Question: "Express the volume of a sphere as a function of its surface area." I know how to do that type of questions, but is there a specific name for that type of questions?
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Expressing something in terms of some other specific thing. That's pretty much it, no special name or anything.
King Squirrel
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It's just a "change of variables."
We have $$V(r)=\frac{4}{3}\pi r^3=\frac{1}{3}r\cdot(\underbrace{4\pi r^2}_{\text{surface area}})=\frac{1}{3}rA=\frac{1}{3}\left(\sqrt{\frac{A}{4\pi}}\right)A$$ where $A=4\pi r^2$ is the surface area.
Usually we express $V$ as a function of the radius $r$, but we have used a "change of variables" to express $V$ as a function of $A$, the surface area.
In general, to express one quantity (say $y$) as a function of another quantity (say $x$) means to write down a relationship $y=f(x)$ where $f(x)$ is some formula that depends only on the variable $x$. Above, you can see that I wrote down $V$ as a function of $A$ because I showed how $V$ depends solely on the value of $A$: $$V=f(A)=\frac{1}{3}\left(\sqrt{\frac{A}{4\pi}}\right)A$$
symplectomorphic
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If one names problems by "how you might solve it" then I agree, but if someone names problems by the overall task, I don't think change of variables is an apt name :s – rschwieb Apr 30 '14 at 03:07
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fair enough, but there is more at stake here -- perhaps contextually, if not literally -- than just "describing a functional relationship," as you put it. there is that, of course, but there is also an illustration of what it means to change the independent variable. (I'm taking for granted the fact that students will naturally think of volume as a function of the radius, not of the surface area.) I was trying to suggest why the problem is significant. – symplectomorphic Apr 30 '14 at 03:10
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Sure, I have to admit that in some cases the method of solution seems inextricable from the problem statement, and that's why I'm torn. I have to push back on your minimization of the task of expressing functional relationships though: it's very fundamental and many students do not grasp the concept! – rschwieb Apr 30 '14 at 10:13
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I'd believe this problem was about that fundamental skill if it didn't start with a quantity that is more typically expressed as a function of something else. – symplectomorphic Apr 30 '14 at 14:07
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I.e., this problem in particular isn't so much about expressing a functional relationship, out of thin air, but of transforming a functional relationship you already know. – symplectomorphic Apr 30 '14 at 14:16
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Dear @symplectomorphic : I don't dispute it is overt and deserves to be discussed. I did my best to tease apart what I feel are the two issues at hand in revisions to my solution. I'm not arguing that either of the two issues takes precedence. Regards (I like your choice of name, btw :) ) – rschwieb Apr 30 '14 at 14:31
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@rschwieb: yes, I don't mean to sound petulant or combative; I was just justifying my sense of what the problem is about. :) – symplectomorphic Apr 30 '14 at 15:20
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You could also express it as $$V(r)=\int_0^r SA(x)\,\mathrm{d}x$$ which is a functional of the surface area :)
Otherwise ignore the above - these problems are called "given B, find A in dependent form"
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@symplectomorphic You're right - see edit :) I think it sort of counts doesn't it?! – Apr 30 '14 at 02:57
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Briefly
The question at its core is about "describing a functional relationship."
You are demonstrating that the volume is a function of the surface area. The quantities following "Write foo in terms of ..." are the things we intend as inputs to the function, and foo is the output of the function.
What about substitution?
Now, one can also note that the most apparent method of solution here involves taking two equations for surface and area in terms of the radius, and then combining them so that the radius quantity disappears, and only the surface and area quantities remain. This makes it tempting to focus on that feature and call it something like a "change of variables" problem.
The drawback is, of course, that a very similar looking problem like "express the surface in terms of the radius" would probably not be considered of the same type, since no "change" is necessary. Both problems are, however, still "express a functional relationship" problems.
A refined explanation
It's unfair to totally ignore the fact that this solution strategy of "changing variables," though. Perhaps a better answer to this question is this:
The question concept type of this problem is to express one quantity as a function of another quantity.
The strategy type for this solution where you combine $S(r)=4\pi r^2$ and $V(r)=\frac43 \pi r^3$ to eliminate $r$ and get your function could be described as "change of variables" or maybe "variable elimination" or "substitution and rearrangement."
In general I don't think it's a good idea to classify problems by their solution types. For one thing, this might trick a student into thinking that a particular problem can only be solved one way, and encourages a "mechanical" approach to problem solving (not desirable!) But to be honest, it probably doesn't hurt that much for students to group problems by solution strategies. It probably works in practice and is a decent first approximation to problem solving.
Classifying solutions by the strategies they employ is very useful and instructive.
Ultimately we want students to have an arsenal of problem solving strategies in their repertoire, but what we don't want is a memorized list of magic bullets.
