HMMT 二月 2000 · 几何 · 第 8 题
HMMT February 2000 — Geometry — Problem 8
题目详情
英文原题
- A sphere is ins rib ed inside a p yramid with a square as a base whose heigh t is timesthe length of one edge of the base. A ub e is ins rib ed inside the sphere. What is the 2
ratio of the v olume of the p yramid to the v olume of the ub e?
解析
英文解析
- By standard form ula, we have that the radius of the ins ib ed ir le, r , is r =
2 a + b
(iso eles triangle that is formed b y utting the p yramid v erti ally in half ( uts the baseq
2 2
b b h
2 2
in to 2 equal re tangles)). h + ( ) = a giv es a = h + . Also sin A = . Therefore,2
2 4 abhr = . Note that the diameter of the ub e is the diameter of the sphere. Let lqb 2
2 h + + b 2
p 4
2 rpbe the length of the side of the ub e, so the diameter of the ub e is l 3 = 2 r , so l = .
p 3
1 25 3
2 3
So, the v olume of the p yramid is b h and the ub e v olume if l . So, the ratio is .
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