Three Spheres in a DishDate: 08/04/99 at 23:18:31 From: Kruger Guy Subject: Radius and spheres Hey you people are great! I always read through the archives, and I have sent in a few problems before. Now I'm a first-year university student and I still find Dr. Math really useful. Anyway, I have this problem that's really bugging me. It was in the Australian Schools national maths competition, so it's supposed to be solved without a calculator. Here's the problem: 3 spheres with radii of 10 cm are inside a hemispherical dish. It is noticed that the tops of all 3 spheres are exactly even with the top of the dish. What is the radius of the dish? I tried to do a diagram but it didn't work. I hope you understand the problem. I'm absolutely stumped for a way to even start this problem, so any help would be greatly appreciated. Thanks heaps and keep up the good work. Kruger Date: 08/05/99 at 12:15:22 From: Doctor Peterson Subject: Re: Radius and spheres Hi, Kruger. A top view will look like this: ********** ****** ****** *** *** ***ooooooooo ooooooooo** **ooo ooo ooo ooo* * o o o o* * o o o o* ** o o o** * o +-------------------+ o * * o \ o / o * * o \ o o / o * * o \ o o / o * * ooo \ oooo*oooo / ooo * * oooooooooo\ | /oooooooooo * * o \ | / o * * o \ | / o * * o \|/ o * ** o + o ** * o o * * o o * ** o o ** *** ooo ooo *** *** ooooooooo ** ****** **** ********** The equilateral triangle has sides 2r, where r = 10 cm is the radius of the small spheres. The distance from the center of any of these spheres to the center of the larger sphere will be 2/3 of the altitude of the triangle. A side view cutting through one of the small spheres will look like this: R *-------ooooooooo--------+----------------------* * ooo ooo / | * * o + |r * * o r / o | * *o / o | * *o +------------+ * * / o * * / r o * + o * *oo ooo * * ooooooooo * * * ** ** ** ** *** *** ****** ****** ******* The horizontal segment D is the same as D just calculated. See where you can go from here. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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