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2003 AMC 12B Problems/Problem 23

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Problem

The number of x-intercepts on the graph of y=\sin(1/x) in the interval (0.0001,0.001) is closest to

\mathrm{(A)}\ 2900\qquad\mathrm{(B)}\ 3000\qquad\mathrm{(C)}\ 3100\qquad\mathrm{(D)}\ 3200\qquad\mathrm{(E)}\ 3300

Solution

The function f(x) = \sin x has roots in the form of \pi n for all integers n. Therefore, we want \frac{1}{x} = \pi n on \frac{1}{10000} \le x \le \frac{1}{1000}, so 1000 \le \frac 1x = \pi n \le 10000. There are \frac{10000-1000}{\pi} \approx \boxed{2900} \Rightarrow \mathrm{(A)} solutions for n on this interval.

See also

2003 AMC 12B (ProblemsResources)
Preceded by
Problem 22
Followed by
Problem 24
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
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