HMMT 二月 2010 · 几何 · 第 7 题

英文解析

7. [ 6 ] You are standing in an infinitely long hallway with sides given by the lines x = 0 and x = 6. Youstart at (3 , 0) and want to get to (3 , 6). Furthermore, at each instant you want your distance to (3 , 6)
   to either decrease or stay the same. What is the area of the set of points that you could pass throughon your journey from (3 , 0) to (3 , 6)?
   √
   21 π
   Answer: 9 3 +
   C2
   A A3
   √
   3 3B
   3 66
   If you draw concentric circles around the destination point, the condition is equivalent to the restriction 0
   that you must always go inwards towards the destination. In the diagram above, the regions throughwhich you might pass are shaded.
   We find the areas of regions A, B, and C separately, and add them up (doubling the area of region A,
   because there are two of them).
   π π π
   The hypotenuse of triangle A is of length 6, and the base is of length 3, so it is a - - triangle
   6 3 2
   √
   √
   9 3
   (30-60-90 triangle) with area . Then the total area of the regions labeled A is 9 3.
   π2
   Since the angle of triangle A nearest the center of the circle (the destination point) is , sector B hasπ 1 2 1 π3
   central angle . Then the area of sector B is r θ = · 36 · = 6 π .
   3 2 2 3
   9 π
   Region C is a half-disc of radius 3, so its area is .
   √2
   21 π
   Thus, the total area is 9 3 + . 2