Difference between revisions of "2000 AMC 8 Problems/Problem 3"
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How many whole numbers lie in the interval between <math>\frac{5}{3}</math> and <math>2\pi</math>? | How many whole numbers lie in the interval between <math>\frac{5}{3}</math> and <math>2\pi</math>? | ||
− | + | <math>\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ \text{infinitely many}</math> | |
− | The smallest whole number in the interval is <math>2</math> because <math>5/3</math> is more than <math>1</math> but less than <math>2</math>. The largest whole number in the interval is <math>6</math> because <math>2\pi</math> is more than <math>6</math> but less than <math>7</math>. There are five whole numbers in the interval. They are <math>2, 3, 4, 5, and 6</math>, so the answer is <math>\boxed{ | + | ==Solution 1 == |
+ | |||
+ | The smallest whole number in the interval is <math>2</math> because <math>5/3</math> is more than <math>1</math> but less than <math>2</math>. The largest whole number in the interval is <math>6</math> because <math>2\pi</math> is more than <math>6</math> but less than <math>7</math>. There are five whole numbers in the interval. They are <math>2</math>, <math>3</math>, <math>4</math>, <math>5</math>, and <math>6</math>, so the answer is <math>\boxed{\text{(D)}\ 5}</math>. | ||
+ | |||
+ | ==Solution 2== | ||
+ | We can approximate <math>2\pi</math> to <math>6</math>. Now we approximate <math>5/3</math> to <math>2</math>. Now we list the integers between <math>2</math> and <math>6</math> including <math>2</math> and <math>6</math>: | ||
+ | <cmath>2,3,4,5,6</cmath> | ||
+ | Hence, the answer is <math>\boxed{D}</math> | ||
==See Also== | ==See Also== | ||
{{AMC8 box|year=2000|num-b=2|num-a=4}} | {{AMC8 box|year=2000|num-b=2|num-a=4}} | ||
+ | {{MAA Notice}} |
Latest revision as of 09:37, 24 July 2024
Contents
Problem
How many whole numbers lie in the interval between and
?
Solution 1
The smallest whole number in the interval is because
is more than
but less than
. The largest whole number in the interval is
because
is more than
but less than
. There are five whole numbers in the interval. They are
,
,
,
, and
, so the answer is
.
Solution 2
We can approximate to
. Now we approximate
to
. Now we list the integers between
and
including
and
:
Hence, the answer is
See Also
2000 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.