# Difference between revisions of "1998 AJHSME Problems/Problem 16"

## Don't Crowd the Isles

Problems 15, 16, and 17 all refer to the following:

In the very center of the Irenic Sea lie the beautiful Nisos Isles. In 1998 the number of people on these islands is only 200, but the population triples every 25 years. Queen Irene has decreed that there must be at least 1.5 square miles for every person living in the Isles. The total area of the Nisos Isles is 24,900 square miles.

### Problem 16

Estimate the year in which the population of Nisos will be approximately 6,000. $\text{(A)}\ 2050 \qquad \text{(B)}\ 2075 \qquad \text{(C)}\ 2100 \qquad \text{(D)}\ 2125 \qquad \text{(E)}\ 2150$

## Solution

### Solution 1

We could triple the population every $25$ years and make a chart:

Year: 2000 Population: 200

Year: 2025 Population: 600

Year: 2050 Population: 1800

Year: 2075 Population: 5400

Year: 2100 Population: 16200

The closest year is 2075, or $\boxed{B}$

### Solution 2

We could find out how many periods of 25 years we need to triple by dividing our total from our present number. $\frac{6000}{200}=30$

The power of $3$ that $30$ is closest to is $27=3^3$

Therefore, after $3$ periods, we will be closest to $6000$. $\boxed{B}$