HMMT 二月 2002 · GEN1 赛 · 第 6 题
HMMT February 2002 — GEN1 Round — Problem 6
题目详情
- A man, standing on a lawn, is wearing a circular sombrero of radius 3 feet. Unfortunately, the hat blocks the sunlight so effectively that the grass directly under it dies instantly. If the man walks in a circle of radius 5 feet, what area of dead grass will result?
解析
- A man, standing on a lawn, is wearing a circular sombrero of radius 3 feet. Unfortu- nately, the hat blocks the sunlight so effectively that the grass directly under it dies instantly. If the man walks in a circle of radius 5 feet, what area of dead grass will result? 2 Solution: 60 π ft Let O be the center of the man’s circular trajectory. The sombrero kills all the grass that is within 3 feet of any point that is 5 feet away from O — i.e. all 2 2 the grass at points P with 2 ≤ OP ≤ 8. The area of this annulus is then π (8 − 2 ) = 60 π square feet.