HMMT 二月 2003 · GEN2 赛 · 第 1 题

英文解析

1. A compact disc has the shape of a circle of diameter 5 inches with a 1-inch-diametercircular hole in the center. Assuming the capacity of the CD is proportional to itsarea, how many inches would need to be added to the outer diameter to double thecapacity?
   Solution: 2
   Doubling the capacity is equivalent to doubling the area, which is initially π [(5 / 2) −2
   (1 / 2) ] = 6 π . Thus we want to achieve an area of 12 π , so if the new diameter is d , we 2
   2 2
   want π [( d/ 2) − (1 / 2) ] = 12 π ⇒ d = 7. Thus we need to add 2 inches to the diameter.