Difference between revisions of "1981 AHSME Problems/Problem 18"
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The number of real solutions to the equation <cmath>\dfrac{x}{100}=\sin x</cmath> is | The number of real solutions to the equation <cmath>\dfrac{x}{100}=\sin x</cmath> is | ||
− | <math>\textbf{(A)}\ 61\qquad\textbf{(B)}\ 62\qquad\textbf{(C)}\ 63\qquad\textbf{(D)}\ 64\qquad\textbf{(E)}\ 65</math> | + | <math>\textbf{(A)}\ 61\qquad\textbf{(B)}\ 62\qquad\textbf{(C)}\ 63\qquad\textbf{(D)}\ 64\qquad\textbf{(E)}\ 65</math> |
+ | ==Solution:== | ||
− | == | + | The answer to this problem is the number of intersections between the graph of <math>f(x) = \sin x</math> and <math>f(x) = \frac{1}{100}x.</math> We can do the right side of the coordinate plane first. Each cycle of the sine wave, consisting of 2π, will have 2 intersections (From the positive part of the sine wave) The line <math>f(x) = \frac{1}{100}x</math> will consist of 16 cycles plus a little bit extra for <math>x</math> from 1 to 100. However, the extra is not complete enough to have any intersection at all. Thus, the number of intersections is <math>2 \cdot 16 = 32.</math> Because of symmetry, we can multiply by two to account for the left side, and subtract one because of the origin. So the answer is <math>32 \cdot 2 - 1 = \textbf{(C)}\ 63.</math> |
− | + | https://www.desmos.com/calculator/z6edqwu1kx - Graph | |
~Eric X | ~Eric X |
Latest revision as of 14:01, 10 November 2022
Problem:
The number of real solutions to the equation is
Solution:
The answer to this problem is the number of intersections between the graph of and
We can do the right side of the coordinate plane first. Each cycle of the sine wave, consisting of 2π, will have 2 intersections (From the positive part of the sine wave) The line
will consist of 16 cycles plus a little bit extra for
from 1 to 100. However, the extra is not complete enough to have any intersection at all. Thus, the number of intersections is
Because of symmetry, we can multiply by two to account for the left side, and subtract one because of the origin. So the answer is
https://www.desmos.com/calculator/z6edqwu1kx - Graph
~Eric X