Saturday, November 12, 2011

another Math League challenge problem!

Try this on for size. It's another of the Fairfax County Math League problems:

I haven't figured this one out yet, but one of my colleagues, who tutors math and science, told me what the approach should be. My attempt at an answer will appear in the comments.



Kevin Kim said...

I strained my brain for a few minutes after class trying to figure this one out... and then my colleague strolled over and told me how easy it was. Think of it this way: all the circles' centers lie on the same line, so you need to think in terms of semicircles.

How do I find the area of the blue region, for example? I need to calculate the area of Circle C, which is the biggest circle. Then I need to calculate the respective areas of Circles A and B. Next, I need to subtract half the area of Circle B from half the area of Circle C, and I also need to add half the area of Circle A.

Then we figure it the other way for red.

Hell, let's do this.

Based on the ratios given, we know the diameter of Circle C is 7. So...

Area of C = (3.5^2)π = 12.25π
Area of B = (2^2)π = 4π
Area of A = (1.5^2)π = 2.25π

Blue region's area:

[(1/2)(12.25π)] - [(1/2)(4π)] + [(1/2)(2.25π)]

= 5.25π

Red region's area

[(1/2)(12.25π)] + [(1/2)(4π)] - [(1/2)(2.25π)]

= 7π

The ratio of blue to red is thus

5.25/7, or 21/28, or 3/4.

That's a lot of work merely to confirm what I had initially suspected! Without doing any math-- and before my colleague had shown me the light-- I had thought the answer would be either 3/4 or 9/16, based purely on the info we were given about the respective diameters of Circles A and B, and my assumptions about the areas of those circles.

So there we are. I'll need to confirm this answer, but I'm pretty sure I'm right.

Charles said...

Looks like the Korean flag went on a bender.

Yep, that's all I've got.

Kevin Kim said...

That's pretty much what I thought when I saw the black-and-white original.

Will be posting the original soon; I brought the sheet home with me from school.

Kevin Kim said...

By the way, 3/4 is confirmed as correct. The original comment I wrote for this problem came to the correct solution, but I had forgotten to calculate area by using radii, not diameters. All my calculations were off by a factor of 2, which turned out not to matter since you lose the 2 once you establish the ratio.