睡莲覆盖池塘需要多久？

How long until the pond is covered in lily pads?

专题: Brainteaser / 脑筋急转弯
难度: L4
来源: QuantQuestion

题目详情

你从一片睡莲开始，池塘其他地方为空。已知睡莲覆盖面积每天翻倍，并且这 1 片睡莲需要 30 天才能覆盖整个池塘。

如果不是从 1 片开始，而是从 8 片（每片增长规律相同）开始，需要多少天才能覆盖池塘？

另一个睡莲问题：池塘里有 27 片睡莲，每片初始面积为 1 平方英尺；池塘总面积为 6000 平方英尺。每片睡莲每天面积翻倍。问多久会覆盖池塘？

You start with a single lily pad sitting on an otherwise empty pond. You are told that the surface area of the lily pad doubles every day and that it will take 30 days for the single lily pad to cover the surface of the pond.

If instead of one lily pad you start with eight lily pads (each identical in characteristics to the original single lily pad), how many days will it take for the surface of the pond to become covered?

Another lily pad problem. There are 27 lily pads on a pond. Each of the lily pads is one square foot in area. The pond is 6,000 square feet in area. Each lily pad doubles its size every day. How long until the pond is covered in lily pads?

解析

(1) 从 8 片开始：8=2^3，相当于比从 1 片开始“提前 3 次翻倍”。

既然从 1 片需要 30 天，则从 8 片需要

30-3=\boxed{27\text{ 天}}.

(2) 27 片、池塘 6000 平方英尺：第 $t$ 天总覆盖面积为 $27\cdot 2^t$ 。

求最小 $t$ 使

27\cdot 2^t\ge 6000\Rightarrow 2^t\ge \frac{6000}{27}\approx 222.22.

因为 $2^7=128$ 、 $2^8=256$ ，所以最小 $t=8$ 。

答案： $\boxed{8\text{ 天}}$ 。