I just gave a solution to this problem, but it didn't get through, apparently. Very annoying. This is a nice problem though, so I think I'll try again.
Neopets Problem Of Week
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Second try: I'm not drawing pictures, so the answer is a bit hard to communicate. The trick to the solution (as is usually the case for problems involving circles) is to look at the lines connecting the centres of the circles and to note that they are composed of the radii of the circles (if the circles are tangent). Let a radius of a small circle be a and the radius of a medium circle be b.
(1) First we find b. Draw a triangle OAB with vertices at the centre of the big circle O (I will refer to circles and their centres by the same name, which is not problematic here) and the centres of two adjacent medium circles A and B. This is an isosceles triangle. The angle AOB (that is the angle at O of the triangle OAB) is 360/10=36 degrees (I'm not sure if that's the american system. I take a full rotation to be 360 degrees). The sides OA=OB=1000+b and AB=2b. Drop a perpendicular from O to the opposing side AB (this bisects O, because OAB is an isosceles triangle). We now see that sin(18)=b/(1000+b)
We solve for b to find:
b=1000 sin(18)/(1-sin(18))
(2) Now we find a
Draw a triangle OPQ with vertices O, P, the centre of a medium circle and Q, the centre of an adjacent small circle. The angle POQ is 18 degrees (to see this, draw another triangle OPR, with R the centre of the other adjacent small circle, and note that ROQ is 36 degrees and the quadrilateral OQPR is symmetric in the line OP). The sides are OP=1000+b, OQ=1000+a, PQ=a+b.
We use the law of cosines to find: (a+b)^2= (1000+a)^2 + (1000+b)^2 - 2(1000+a)(1000+b)cos(18). We solve for a to find:
a=(1000000 + 1000b)(1-cos(18))/(b + (1000 + b)cos(18) -1000)
(3) The requested area Z is now straightforward:
Z=10*pi*a^2
My calculations give:
b=447, a=86, Z=232352
(There may be significant rounding erors in the final answers, because I didn't feel like punching in the entire calculations in my pocket calculator, without rounding the answers in between.)
By the way: is neopets what I think it is? I didn't think it had anything to do with geometry.
Neopets is a site with little pets and it really doesn't have anything to do with math, but occasionally they come up with an entertaining math problem.
I used the law of sines to get the brown radius, then the theorem of the Kissing Circles to get the blue radius.
Well, I had never heard of that theorem (in fact I hadn't even heard of either the law of sines or of cosines. I just found the law of cosines when I recognized my results and googled "generalized Pythagoras theorem"). Fortunately, it seems our solutions coincide.
The theorem of Kissing Circles does rather simplify the problem, but it's not very interesting anymore; it's more or less a fill in exercise like "find the third side" using the Pythaghoras theorem.
Week 352 Lenny Conundrum:
There is a circle O (yellow), radius 1000. There are 10 congruent medium sized circles (brown) on the exterior of O, all tangent to O. They are each tangent to two others as well. There are 10 more small circles (blue), all congruent, tangent to O, as well as 2 medium sized circles. What is the total area covered by all of the small circles?
In other words, the yellow circle is of radius 1000, what is the area shaded in blue?