2003 AMC 12B Problem 23

Attempt Problem 23 of the 2003 AMC 12B below, then check your answer against the professionally curated solution 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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23.

The number of

x

-intercepts on the graph of

y = \sin(\frac{1}{x})

in the interval

(0.0001, 0.001)

is closest to

2900

3000

3100

3200

3300

Answer: A

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

Difficulty rating: 1950

Small hint:

\sin(\frac{1}{x}) = 0

when

\frac{1}{x} = k\pi

for a nonzero integer

k,

i.e.

x = \dfrac{1}{k\pi}

Big hint:

Count integers

k

with

\dfrac{1000}{\pi} \lt k \lt \dfrac{10000}{\pi}

Solution:

The intercepts occur where

\frac{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

\begin{aligned} &\left\lfloor \frac{10000}{\pi} \right\rfloor - \left\lfloor \frac{1000}{\pi} \right\rfloor \\ &= 3183 - 318 = 2865, \end{aligned}

closest to

2900.

Thus, the correct answer is A.

Problem 23 in other years