HMMT 二月 2003 · CALC 赛 · 第 8 题

HMMT February 2003 — CALC Round — Problem 8

A right circular cone with a height of 12 inches and a base radius of 3 inches is filled with water and held with its vertex pointing downward. Water flows out through a hole at the vertex at a rate in cubic inches per second numerically equal to the height of the water in the cone. (For example, when the height of the water in the cone is 4 inches, water flows out at a rate of 4 cubic inches per second.) Determine how many seconds it will take for all of the water to flow out of the cone.

解析

A right circular cone with a height of 12 inches and a base radius of 3 inches is filled with water and held with its vertex pointing downward. Water flows out through a hole at the vertex at a rate in cubic inches per second numerically equal to the height of the water in the cone. (For example, when the height of the water in the cone is 4 inches, water flows out at a rate of 4 cubic inches per second.) Determine how many seconds it will take for all of the water to flow out of the cone. Solution: 9 π/ 2 When the water in the cone is h inches high, it forms a cone similar to the original, so 3 that its base has radius h/ 4 and its volume is hence πh / 48. The given condition then states that ( ) 3 2 d πh πh dh dh 32 = − h ⇒ · = − h ⇒ 2 h · = − . dt 48 16 dt dt π 2 Integrating with respect to t , we get that h = − 32 t/π + C ; setting t = 0, h = 12, we get C = 144. The cone empties when h = 0, so 0 = − 32 t/π + 144 ⇒ t = 9 π/ 2. 2