HMMT 二月 2019 · 冲刺赛 · 第 27 题

HMMT February 2019 — Guts Round — Problem 27

[ 15 ] Consider the eighth-sphere { ( x, y, z ) | x, y, z ≥ 0 , x + y + z = 1 } . What is the area of its projection onto the plane x + y + z = 1?

解析

[ 15 ] Consider the eighth-sphere { ( x, y, z ) | x, y, z ≥ 0 , x + y + z = 1 } . What is the area of its projection onto the plane x + y + z = 1? Proposed by: Yuan Yao √ π 3 Answer: 4 Consider the three flat faces of the eighth- ball . Each of these is a quarter-circle of radius 1 , so each has π area . Furthermore, the projections of these faces cover the desired area without overlap. To find the 4 projection factor one can find the cosine of the angle θ between the planes, which is the same as the angle between their normal vectors. Using the dot product formula for the cosine of the angle between (1 , 0 , 0) · (1 , 1 , 1) 1 1 √ √ two vectors, cos θ = = . Therefore, each area is multiplied by by the projection, | (1 , 0 , 0) || (1 , 1 , 1) | 3 3 √ π 1 π 3 √ so the area of the projection is 3 · · = . 4 4 3