HMMT 二月 2000 · 几何 · 第 8 题

HMMT February 2000 — Geometry — Problem 8

A sphere is ins rib ed inside a p yramid with a square as a base whose heigh t is times 2 the length of one edge of the base. A ub e is ins rib ed inside the sphere. What is the ratio of the v olume of the p yramid to the v olume of the ub e?

解析

By standard form ula, w e ha v e 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 base q 2 2 b b h 2 2 2 in to 2 equal re tangles)). h + ( ) = a giv es a = h + . Also sin A = . Therefore, 2 4 a bh q r = . Note that the diameter of the ub e is the diameter of the sphere. Let l 2 b 2 2 h + + b 4 p 2 r p b e the length of the side of the ub e, so the diameter of the ub e is l 3 = 2 r , so l = . 3 p 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 . 3 6