Wednesday, April 11, 2012

The Third Wheel...

Two wheels with  diameters 18 and 8 are touching and are on level ground. Show that the diameter of a 3rd wheel on the ground which touches the other 2 is 2.88.


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7 comments:

Pat B said...

The Kiss Precise:
http://pballew.net/soddy.html

Dave Marain said...

Beautiful, Pat! I can always count on you for a detailed history lesson!

Maybe I should have posted osculating circle problems on 2-14...

I was attempting to concretize the problem by placing wheels on concrete to entice the student to give it a 'whirl'. Just my little spin on the problem...

"And the world goes round and round and round in the circle game"

Justin Aion said...

It solved it using pythagorean theorem, but doing so lead me to believe that there was a more efficient way to do it.

Justin Aion said...

Trig functions, not Pythagorean theorem...

I'm spending WAY too much time teaching addition...

Dave Marain said...

Justin,
I also find that trig often helps but I too will get that annoying feeling that Euclid is turning over in his ____.

Anyway, one can use Pyth to show that the length of the common tang segm for 2 ext tang circles is 2√(Rr). This allows us to set up an eqn for the middle radius.

BTW, send me an email and bring me up to date on your life. I can tell there have been many changes...

Ed Bujak said...

This problem is a Ford circle:

http://en.wikipedia.org/wiki/Ford_circle

1/SQROOT(Rmid) = 1/SQROOT(Rleft) + 1/SQROOT(Rright)
where:
Rleft = 4
Rright = 9
Therefore:
Rmid = 36/25 = 1.44

and hence the diameter is 2.88

How did Ford circle equation come about? Can others post their solution(s)?

Dave Marain said...

Thanks, Ed! These geom problems of antiquity continue to provide a treasure trove of problems for teachers. Even more intriguing is the Apollonian Gasket in which the 3 circles are tangent to a common circle, leading to a famous fractal.

Let R3 denote the rad of middle circle.
As I indicated to Justin, length of common tang segm for circles with radii R1 and R3 can be shown in a separate derivation to be 2√(R1•R3). Similarly, length of tang segm for circles R3 and R2 is 2√(R2•R3). Thus, 2√(R1•R3) + 2√(R2•R3) = 2√(R1•R2)
Now divide both sides by 2√(R1•R2•R3) to obtain the Ford equation.

Keep sharing your knowledge and ideas!