HMMT 二月 2004 · 几何 · 第 3 题

HMMT February 2004 — Geometry — Problem 3

A swimming pool is in the shape of a circle with diameter 60 ft. The depth varies linearly along the east-west direction from 3 ft at the shallow end in the east to 15 ft at the diving end in the west (this is so that divers look impressive against the sunset) but does not vary at all along the north-south direction. What is the volume of the 3 pool, in ft ?

解析

A swimming pool is in the shape of a circle with diameter 60 ft. The depth varies linearly along the east-west direction from 3 ft at the shallow end in the east to 15 ft at the diving end in the west (this is so that divers look impressive against the sunset) but does not vary at all along the north-south direction. What is the volume of the 3 pool, in ft ? Solution: 8100 π Take another copy of the pool, turn it upside-down, and put the two together to form 2 a cylinder. It has height 18 ft and radius 30 ft, so the volume is π (30 ft) · 18 ft = 3 3 16200 π ft ; since our pool is only half of that, the answer is 8100 π ft .