What Type of Question is this??

Question

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?

Answer 1

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.

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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.

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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.

Answer 2

Expressing something in terms of some other specific thing. That's pretty much it, no special name or anything.

Answer 3

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 $A$: $$V=f(A)=\frac{1}{3}\left(\sqrt{\frac{A}{4\pi}}\right)A$$

Answer 4

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"