pizer@ecsvax.UUCP (07/15/84)
References: <ecsvax.2878>,<decwrl.2523> The following is the answer to the second geometry problem: #2 >>This time, you have 5 circles, each externally tangent to the one next to it >>and decreasing in size. Each circle is also externally tangent to lines L1 >>and L2, which are not parallel. If this is difficult to picture, imagine L1 >>and L2 being a funnel, with 5 marbles caught inside, each one touching the >>funnel, and the one(s) next to it. Anyway you are given that the radius of >>the large circle is 18, the radius of the small circle is 8, and you are to >>find the radius of the circle in the center. HINT: The answer is *NOT* 13! >>Good luck! In this problem, it appeared fairly obvious that the answer would either be the arithmetic or geometric mean. Since I told you it wasn't 13, logic would indicate the answer to be twelve. My answer involved setting up proportions and then solving for a variable. A more complete answer was recently posted by Van Luong Nguyen <decwrl.2523> explaining how since the sequence is simply a geometric progression, any circle's radii can be calculated in the sequence. His explanation will also save me the trouble and space of going through and solving it now. Several other methods were mailed to me, but they all involved setting up different proportions. Since this is the shortest solution, I'll stick this here: Credit must also be given to the people who got these questions together. All of the problems came off of this year's and last year's American High School Math Exam (AHSME), sponsored by the Mathematical Association of America, Society of Actuaries, Mu Alpha Theta, National Council of Teachers of Mathematics, and Casualty Actuarial Society. This exam is a qualifying exam for the American Invitational Mathematics Exam (AIME). Questions about ordering problem books can be directed to: Prof. Walter E. Mientka, Executive Director MAA Committee on High School Contests Department of Mathematics and Statistics University of Nebraska Lincoln, NE 68588-0322 Billy Pizer ({decvax,akgua,ihnp4,burl}!mcnc!ecsvax!pizer) -- "Why is air speed different from ground speed?" "Because the Earth is round and the air is flat." --