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

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==Don't Crowd the Isles==
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==Don't Crowd the Isles==
  
 
Problems 15, 16, and 17 all refer to the following:
 
Problems 15, 16, and 17 all refer to the following:
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<math>\text{(A)}\ 2050 \qquad \text{(B)}\ 2075 \qquad \text{(C)}\ 2100 \qquad \text{(D)}\ 2125 \qquad \text{(E)}\ 2150</math>
 
<math>\text{(A)}\ 2050 \qquad \text{(B)}\ 2075 \qquad \text{(C)}\ 2100 \qquad \text{(D)}\ 2125 \qquad \text{(E)}\ 2150</math>
  
==Solution 1==
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==Solution==
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===Solution 1===
  
We could triple the population every <math>25</math> years and add them up.
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We could triple the population every <math>25</math> years and make a chart:
  
 
Year: 2000
 
Year: 2000
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Year: 2050
 
Year: 2050
Population:1800
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Population: 1800
  
 
Year: 2075
 
Year: 2075
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Population: 16200
 
Population: 16200
  
The closest is Year 2075, or <math>\boxed{B}</math>
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The closest year is 2075, or <math>\boxed{B}</math>
  
==Solution 2==
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===Solution 2===
  
 
We could find out how many periods of 25 years we need to triple by dividing our total from our present number.
 
We could find out how many periods of 25 years we need to triple by dividing our total from our present number.
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Therefore, after <math>3</math> periods, we will be closest to <math>6000</math>.
 
Therefore, after <math>3</math> periods, we will be closest to <math>6000</math>.
  
<math>boxed{B}</math>
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<math>\boxed{B}</math>
  
  
 
== See also ==
 
== See also ==
{{AJHSME box|year=1998|before=[[1997 AJHSME Problems|1997 AJHSME]]|after=[[1999 AMC 8 Problems|1999 AMC 8]]}}
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{{AJHSME box|year=1998|num-b=15|num-a=17}}
 
* [[AJHSME]]
 
* [[AJHSME]]
 
* [[AJHSME Problems and Solutions]]
 
* [[AJHSME Problems and Solutions]]
 
* [[Mathematics competition resources]]
 
* [[Mathematics competition resources]]
 +
{{MAA Notice}}

Latest revision as of 19:56, 31 October 2016

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 $6000$.

$\boxed{B}$


See also

1998 AJHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 15
Followed by
Problem 17
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

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