2003 AMC 12B Problem 23

Below is the professionally curated solution for Problem 23 of the 2003 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2003 AMC 12B solutions, or check the answer key.

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Concepts:counting integers in a range floor and ceiling functions trigonometry

Difficulty rating: 1950

23.

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

$2900$

$3000$

$3100$

$3200$

$3300$

Solution:

The intercepts occur where $1/x = k\pi,$ that is $x = \dfrac{1}{k\pi}$ for a nonzero integer $k.$

The condition $0.0001 \lt \dfrac{1}{k\pi} \lt 0.001$ becomes $\frac{1000}{\pi} \lt k \lt \frac{10000}{\pi}.$

The number of such integers is $\left\lfloor \frac{10000}{\pi} \right\rfloor - \left\lfloor \frac{1000}{\pi} \right\rfloor = 3183 - 318 = 2865,$ closest to $2900.$

Thus, the correct answer is A.

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