   SEARCH HOME Math Central Quandaries & 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.     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.