HMMT 二月 2002 · 冲刺赛 · 第 30 题

英文解析

30. A conical flask contains some water. When the flask is oriented so that its base ishorizontal and lies at the bottom (so that the vertex is at the top), the water is 1 inch deep.
    When the flask is turned upside-down, so that the vertex is at the bottom, the water is 2
    inches deep. What is the height of the cone?
    √7
    1 93
    Solution: + . Let h be the height, and let V be such that V h equals the volume 3
    2 6
    of the flask. When the base is at the bottom, the portion of the flask not occupied by waterforms a cone similar to the entire flask, with a height of h − 1; thus its volume is V ( h − 1) .3
    When the base is at the top, the water occupies a cone with a height of 2, so its volume is
    V · 2 . Since the water’s volume does not change,3
    3 3
    V h − V ( h − 1) = 8 V
    2 3 3
    ⇒ 3 h − 3 h + 1 = h − ( h − 1) = 8
    ⇒ 3 h − 3 h − 7 = 0 .2
    √
    1 93
    Solving via the quadratic formula and taking the positive root gives h = + .
    2 6