HMMT 二月 1998 · CALC 赛 · 第 2 题
HMMT February 1998 — CALC Round — Problem 2
题目详情
英文原题
Question Two . [3 points]
A cube with sides 1 m in length is filled with water, and has a tinyhole through which the water drains into a cylinder of radius 1 m.
If the water level in the cube is falling at a rate of 1 , at whatcmsrate is the water level in the cylinder rising?
解析
英文解析
- Problem: A cube with sides 1 m in length is filled with water, and has a tiny hole through which thewater drains into a cylinder of radius 1 m. If the water level in the cube is falling at a rate of 1 cm/s, at whatrate is the water level in the cylinder rising?
Solution: The magnitude of the change in volume per unit time of the two solids is the same. The changein volume per unit time of the cube is 1 cm · m /s. The change in volume per unit time of the cylinder is 2
dh dhπ · · m , where is the rate at which the water level in the cylinder is rising.2
dt dtdh 1
2 2
Solving the equation π · · m = 1 cm · m /s yields cm/s .
dt π 2