AMC 8 1998 Problem 17
AMC8 1998 年第 17 题
数列与规律★★★☆☆
In how many years, approximately, from 1998 will the population of Nisos be as much as Queen Irene has proclaimed that the islands can support?
在 Irenic 海的中央坐落着美丽的 Nisos 群岛。1998 年这些岛屿上的人口仅为 200，但人口每 25 年翻三倍。Irene 女王已颁布法令，岛上每个人必须有至少 1.5 平方英里的空间。Nisos 群岛的总面积为 24,900 平方英里。
大约从 1998 年起多少年后，Nisos 的人口会达到 Irene 女王宣布的群岛可承载的数量？
- A.
- B.
- C.
- D.
- E.
正确答案 · Correct Answer
C
解析 · Solution
Solution 1
We can divide the total area by how much will be occupied per person:
can stay on the island at its maximum capacity.
We can divide 16600 by the current population in 1998 which is 200 to see by what factor the population increases:
-fold increase in population.
Thus, the population increases by a factor 83. This is very close to , and so there are about 4 triplings of the island's population.
It takes 4×25=100=C years to triple the island's population four times in succession.
We can continue the pattern, and because the pattern increases numbers rapidly, it won't be hard. can live on the island at its maximum capacity.
1998: 200 people; 2023: 600 people; 2048: 1800 people; 2073: 5400 people; 2098: 16200 people.
After the year 2098, it will not be possible for the next increase to occur, because when 16200 is tripled, it is way more than the maximum capacity. Thus, the answer is 2098-1998 = 100, or .
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