Difference between revisions of "2020 AMC 12A Problems/Problem 9"
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~MRENTHUSIASM ~lopkiloinm ~hi13 ~annabelle0913 ~codecow | ~MRENTHUSIASM ~lopkiloinm ~hi13 ~annabelle0913 ~codecow | ||
+ | |||
+ | ==Video Solution== | ||
+ | https://youtu.be/ZdVut0V1O4g with tips on graphing trig functions fast ~MathProblemSolvingSkills.com | ||
==See Also== | ==See Also== |
Latest revision as of 04:07, 8 July 2022
Contents
Problem
How many solutions does the equation have on the interval
Solution
We count the intersections of the graphs of and
- The graph of
has a period of
asymptotes at
and zeros at
for some integer
On the interval
the graph has five branches:
Note that
for the first branch,
for the three middle branches, and
for the last branch. Moreover, all branches are strictly increasing.
- The graph of
has a period of
and zeros at
for some integer
On the interval
note that
Moreover, the graph is strictly decreasing.
The graphs of and
intersect once on each of the five branches of
as shown below:
Therefore, the answer is
~MRENTHUSIASM ~lopkiloinm ~hi13 ~annabelle0913 ~codecow
Video Solution
https://youtu.be/ZdVut0V1O4g with tips on graphing trig functions fast ~MathProblemSolvingSkills.com
See Also
2020 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 8 |
Followed by Problem 10 |
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 AMC 12 Problems and Solutions |
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