Difference between revisions of "2021 AMC 10B Problems/Problem 1"
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<math>\textbf{(A)} ~9 \qquad\textbf{(B)} ~10 \qquad\textbf{(C)} ~18 \qquad\textbf{(D)} ~19 \qquad\textbf{(E)} ~20</math> | <math>\textbf{(A)} ~9 \qquad\textbf{(B)} ~10 \qquad\textbf{(C)} ~18 \qquad\textbf{(D)} ~19 \qquad\textbf{(E)} ~20</math> | ||
− | ==Solution== | + | ==Solution 1== |
Since <math>3\pi</math> is about <math>9.42</math>, we multiply 9 by 2 and add 1 to get <math> \boxed{\textbf{(D)}\ ~19} </math>~smarty101 | Since <math>3\pi</math> is about <math>9.42</math>, we multiply 9 by 2 and add 1 to get <math> \boxed{\textbf{(D)}\ ~19} </math>~smarty101 | ||
Revision as of 01:45, 12 February 2021
Contents
[hide]Problem
How many integer values of satisfy
?
Solution 1
Since is about
, we multiply 9 by 2 and add 1 to get
~smarty101
Solution 2
There are two cases here.
When and
So then
When and
So then
. Dividing by
and flipping the sign, we get
From case 1 and 2, we need . Since
is an integer, we must have
between
and
. There are a total of
-PureSwag
Solution 3
. Since
is approximately
,
is approximately
. We are trying to solve for
, where
. Hence,
, for
. The number of integer values of
is
. Therefore, the answer is
.
~ {TSun} ~
2021 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by First Problem |
Followed by Problem 2 | |
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All AMC 10 Problems and Solutions |