Difference between revisions of "2020 AMC 12A Problems/Problem 9"
MRENTHUSIASM (talk | contribs) m (→Solution) |
|||
Line 106: | Line 106: | ||
~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== |
Revision as of 17:11, 7 July 2022
Contents
[hide]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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.