HMMT 二月 2006 · 冲刺赛 · 第 28 题

英文解析

28. A pebble is shaped as the intersection of a cube of side length 1 with the solid spheretangent to all of the cube’s edges. What is the surface area of this pebble?
    √
    6 2 − 5
    Answer: π
    Solution: Imagine drawing the sphere and the cube. Take a cross section, with a 2
    plane parallel to two of the cube’s faces, passing through the sphere’s center. In thiscross section, the sphere looks like a circle, and the cube looks like a square (of sidelength 1) inscribed in that circle. We can now calculate that the sphere has diameter
    √9
    d := 2 and surface area S := πd = 2 π , and that the sphere protrudes a distance of 2
    √
    2 − 1
    x := out from any given face of the cube.
    It is known that the surface area chopped off from a sphere by any plane is proportional 2
    to the perpendicular distance thus chopped off. Thus, each face of the cube chops ofxa fraction of the sphere’s surface. The surface area of the pebble contributed by thedx 1
    sphere is thus S · (1 − 6 · ), whereas the cube contributes 6 circles of radius , withd 2
    ( )
    1 32
    total area 6 · π = π . The pebble’s surface area is therefore
    2 2
    ( )
    √ √
    ( )
    x 3 2 − 1 3 6 2 − 5
    S · 1 − 6 · + π = 2 π · 1 − 6 · √ + π = π.
    d 2 2 2
    2 2
    2 2