Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1998 AJHSME Problems/Problem 17"

(Added navigational box)
(→‎Solution)
Line 15: Line 15:
 
==Solution==
 
==Solution==
  
−
We can divide the total area by how much will be occupied per person.
+
We can divide the total area by how much will be occupied per person:
  
−
<math>\frac{24900}{1.5}=16600</math>
+
<math>\frac{24900 \text{ acres}}{1.5 \text{ acres/person}}=16600 \text{ people}</math> can stay on the island at its maximum capacity.
  
−
We can divide the total amount of people by how many in the year <math>2000</math>
+
We can divide this by the current populationon the island in the year <math>1998</math> to see by what factor the population increases:
  
−
<math>\frac{16600}{200}=83</math>
+
<math>\frac{16600}{200}=83</math>-fold increase in population.
  
−
<math>83</math>'s closest power of <math>3</math> is <math>81=3^4</math>
+
Thus, the population increases by a factor <math>83</math>.  This is very close to <math>3 \times 3 \times 3 \times 3 = 81</math>, and so there are about <math>4</math> triplings of the island's population.
  
−
<math>4\times25=100=boxed{C}</math>
+
It takes <math>4\times25=100=\boxed{C}</math> years to triple the island's population four times in succession.
  
 
== See also ==
 
== See also ==

Revision as of 10:55, 31 July 2011

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

We can divide the total area by how much will be occupied per person:

$\frac{24900 \text{ acres}}{1.5 \text{ acres/person}}=16600 \text{ people}$ can stay on the island at its maximum capacity.

We can divide this by the current populationon the island in the year $1998$ 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.

See also

1998 AJHSME (Problems • Answer Key • Resources)
Preceded by
1997 AJHSME 		Followed by
1999 AMC 8
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AJHSME/AMC 8 Problems and Solutions
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1998_AJHSME_Problems/Problem_17&oldid=41017"