# 1998 AJHSME Problems/Problem 17

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## 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 17

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? $\text{(A)}\ 50\text{ yrs.} \qquad \text{(B)}\ 75\text{ yrs.} \qquad \text{(C)}\ 100\text{ yrs.} \qquad \text{(D)}\ 125\text{ yrs.} \qquad \text{(E)}\ 150\text{ yrs.}$

## Solution 1

We can divide the total area by how much will be occupied per person: $\frac{24900 \text{ acres}}{1.5 \text{ acres per person}}=16600 \text{ people}$ 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: $\frac{16600}{200}=83$-fold increase in population.

Thus, the population increases by a factor $83$. This is very close to $3 \times 3 \times 3 \times 3 = 81$, and so there are about $4$ triplings of the island's population.

It takes $4\times25=100=\boxed{C}$ years to triple the island's population four times in succession.

## Solution 2

We can continue the pattern, and because the pattern increases numbers rapidly, it won't be hard. $\frac{24900 \text{ acres}}{1.5 \text{ acres per person}}=16600 \text{ people}$ can live on the island at its maximum capacity. $1998: 200 \text{ people}$ $2023: 600 \text{ people}$ $2048: 1800 \text{ people}$ $2073: 5400 \text{ people}$ $2098: 16200 \text{ people}$
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 $\boxed{C}$

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