Difference between revisions of "2021 AMC 12B Problems/Problem 13"
(→Solution 2: wrong answer choice) |
MRENTHUSIASM (talk | contribs) m (Made the final answer's boxes more consistent.) |
||
Line 4: | Line 4: | ||
==Solution 1== | ==Solution 1== | ||
− | First, move terms to get <math>1+5\cos 3x=3\sin x</math>. After graphing, we find that there are <math>\boxed{6}</math> solutions (two in each period of <math>5\cos 3x</math>). -dstanz5 | + | First, move terms to get <math>1+5\cos 3x=3\sin x</math>. After graphing, we find that there are <math>\boxed{\textbf{(D) }6}</math> solutions (two in each period of <math>5\cos 3x</math>). -dstanz5 |
− | |||
==Solution 2== | ==Solution 2== | ||
Line 25: | Line 24: | ||
xaxis("$x$",BottomTop,LeftTicks); | xaxis("$x$",BottomTop,LeftTicks); | ||
yaxis("$y$",LeftRight,RightTicks(trailingzero)); | yaxis("$y$",LeftRight,RightTicks(trailingzero)); | ||
− | |||
− | |||
add(legend(),point(E),20E,UnFill); | add(legend(),point(E),20E,UnFill); | ||
</asy> | </asy> | ||
− | We have <math>\boxed{(D) 6}</math> solutions. ~Jamess2022 (burntTacos) | + | We have <math>\boxed{\textbf{(D) }6}</math> solutions. ~Jamess2022 (burntTacos) |
== Video Solution by OmegaLearn (Using Sine and Cosine Graph) == | == Video Solution by OmegaLearn (Using Sine and Cosine Graph) == |
Revision as of 18:47, 24 June 2021
Contents
[hide]Problem
How many values of in the interval satisfy
Solution 1
First, move terms to get . After graphing, we find that there are solutions (two in each period of ). -dstanz5
Solution 2
We can graph two functions in this case: and . Using transformation of functions, we know that is just a cos function with amplitude 5 and period . Similarly, is just a sin function with amplitude 3 and shifted 1 unit downwards. So: We have solutions. ~Jamess2022 (burntTacos)
Video Solution by OmegaLearn (Using Sine and Cosine Graph)
~ pi_is_3.14
Video Solution by Hawk Math
https://www.youtube.com/watch?v=p4iCAZRUESs
See Also
2021 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 12 |
Followed by Problem 14 |
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.