As fun as Richey Numbers are, it's time to move on.

I've come up with an unusual puzzle to get me back on the random-math track. It's a problem that stumped me through all of 10th grade English class, the year I formally learned of Polar Graphing. I was curious about the graph of r = Θ, which takes the form of a spiral. On the interval [0, π/2] the graph hits a maximum x value. We have the points (x,y) = (0,0) and (0, π/2), but in between there the graph is an unusual curve in the first quadrant. I asked, what was the maximum x value the graph hits in that interval?

I had no idea how to solve it at first. I asked numerous math teachers, and those who didn't completely ignore me weren't sure what to do. This year I'm in Calculus AB, and in light of a recent unit regarding Optimization I was struck with an idea.

We have the relationships between Polar and Cartesian coordinates that x^2+y^2 = r^2, and tan Θ = y/x. The function says that r = Θ, so

x^2 + y^2 = Θ^2

Also,

xtan Θ = y => x^2 tan^2(Θ) = y^2, so

x^2 + x^2 tan^2 (Θ) = Θ^2

x^2(1+tan^2 (Θ)) = Θ^2

Since 1+tan^2 (Θ) = sec^2 (Θ),

x^2 = Θ^2 cos^2 (Θ)

x = Θ cosΘ

Now we have a simple relationship relating x and theta. All we need to do is differentiate and find where the maximums and minimums occur on Θ ε [0, π/2].

dx/dΘ = cos Θ - Θ sin Θ

Θ = cot Θ, a problem best left to the calculator. Approximately, Θ = .86033, and x = Θ cos Θ = .56110.

The maximums in x and y in any interval can be found using the same method; they are the solutions to the equations Θ = cot Θ and Θ = tan Θ, respectively. I have yet to try this out on other polar functions.