HMMT 二月 2004 · 冲刺赛 · 第 18 题

HMMT February 2004 — Guts Round — Problem 18

[8] On a spherical planet with diameter 10 , 000 km, powerful explosives are placed at the north and south poles. The explosives are designed to vaporize all matter within 5 , 000 km of ground zero and leave anything beyond 5 , 000 km untouched. After the explosives are set off, what is the new surface area of the planet, in square kilometers? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HARVARD-MIT MATHEMATICS TOURNAMENT, FEBRUARY 28, 2004 — GUTS ROUND

解析

On a spherical planet with diameter 10 , 000 km, powerful explosives are placed at the north and south poles. The explosives are designed to vaporize all matter within 5 , 000 km of ground zero and leave anything beyond 5 , 000 km untouched. After the explosives are set off, what is the new surface area of the planet, in square kilometers? Solution: 100 , 000 , 000 π The explosives have the same radius as the planet, so the surface area of the “cap” removed is the same as the new surface area revealed in the resulting “dimple.” Thus 2 the area is preserved by the explosion and remains π · (10 , 000) .