Math CentralQuandaries & Queries


Question from Gilligan, a student:

The volume V of a right circular cone is V = (1/3)pi(r^2)(h).

If the height is TWICE the radius, express the volume V as a function of r.

NOTE: I see the phrase "express a certain variable as a function of another variable" a lot in math books. What exactly does that mean? Is there another way to say the same thing?

Hi Gilligan.

When we ask you to "express a certain variable as a function of another variable" we mean to construct a function such that you can feed one variable into it and get the other variable.

y = 2x - 5

y = f(x) = 2x - 5

What I wrote above is expressing y as a function of x. I could say, "Change that to express x as a function of y" and you would transform it into the equivalent equation:

x = (y + 5) / 2

So "express the volume V as a function of r" means get V by itself on the left side of the equal sign and r (and no other variables) in an expression on the right side of the equal sign.

Here's another example. The Area of a circle is A = π. The Circumference of a circle is C = 2πr. Express the Area A as a function of the Circumference C.

This means provide an equation with just A in it and C in it, where A is the only thing on the left hand side and there are no variables other than C on the right side. Here's how I'd solve this:

C = 2πr, therefore r = C / (2π). Substitute this into A = πr² and I have A = π[ C / (2π) ]². This simplifies to A = C² / (4π). That's the answer.

Now you should be able to tackle your question.
Stephen La Rocque.

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