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2003 AMC 12B Problems/Problem 23 - Revision history
2024-03-28T18:35:13Z
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MRENTHUSIASM: Undo revision 189361 by Lkarhat (talk)
2023-02-09T22:39:28Z
<p>Undo revision 189361 by <a href="/wiki/index.php/Special:Contributions/Lkarhat" title="Special:Contributions/Lkarhat">Lkarhat</a> (<a href="/wiki/index.php?title=User_talk:Lkarhat&action=edit&redlink=1" class="new" title="User talk:Lkarhat (page does not exist)">talk</a>)</p>
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MRENTHUSIASM
https://artofproblemsolving.com/wiki/index.php?title=2003_AMC_12B_Problems/Problem_23&diff=189361&oldid=prev
Lkarhat: I LIKE TURTLES AND MAILBOXES!!!
2023-02-09T21:17:32Z
<p>I LIKE TURTLES AND MAILBOXES!!!</p>
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Lkarhat
https://artofproblemsolving.com/wiki/index.php?title=2003_AMC_12B_Problems/Problem_23&diff=177827&oldid=prev
Isabelchen: /* Solution */
2022-09-01T14:17:45Z
<p><span dir="auto"><span class="autocomment">Solution</span></span></p>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Solution==</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Solution==</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>We know that <math>x</math> belongs to the interval <math>(0.0001,0.001)</math> for <math>sin(1/x)</math>. We see that when we plug in <math>x</math> into <math>sin(1/x)</math>, the argument <math>(1/x)</math> is always from the range <math>(1000, 10000)</math>. Therefore, the problem simply asks for all the zeros of <math>sin(x)</math> with <math>x</math> values between <math>(1000, 10000)</math>. We know that the <math>x</math> values of any sine graph is <math>\pi(n-1)</math> so, we see that values of <math>n</math> are any integer value from <math>320</math> to <math>3184</math> and therefore gives us an answer of approximately <math>2865</math> which is answer <math>\boxed{\text{(A)} 2900}</math></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>We know that <math>x</math> belongs to the interval <math>(0.0001,0.001)</math> for <math><ins class="diffchange diffchange-inline">\</ins>sin(1/x)</math>. We see that when we plug in <math>x</math> into <math><ins class="diffchange diffchange-inline">\</ins>sin(1/x)</math>, the argument <math>(1/x)</math> is always from the range <math>(1000, 10000)</math>. Therefore, the problem simply asks for all the zeros of <math><ins class="diffchange diffchange-inline">\</ins>sin(x)</math> with <math>x</math> values between <math>(1000, 10000)</math>. We know that the <math>x</math> values of any sine graph is <math>\pi(n-1)</math> so, we see that values of <math>n</math> are any integer value from <math>320</math> to <math>3184</math> and therefore gives us an answer of approximately <math>2865</math> which is answer <math>\boxed{\text{(A)} 2900}</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>~Jske25</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>~Jske25</div></td></tr>
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Isabelchen
https://artofproblemsolving.com/wiki/index.php?title=2003_AMC_12B_Problems/Problem_23&diff=151480&oldid=prev
Jske25 at 08:17, 14 April 2021
2021-04-14T08:17:34Z
<p></p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 08:17, 14 April 2021</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Solution ==</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Solution ==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The function <math>f(x) = \sin x</math> has roots in the form of <math>\pi n</math> for all integers <math>n</math>. Therefore, we want <math>\frac{1}{x} = \pi n</math> on <math>\frac{1}{10000} \le x \le \frac{1}{1000}</math>, so <math>1000 \le \frac 1x = \pi n \le 10000</math>. There are <math>\frac{10000-1000}{\pi} \approx \boxed{2900} \Rightarrow \mathrm{(A)}</math> solutions for <math>n</math> on this interval.  </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The function <math>f(x) = \sin x</math> has roots in the form of <math>\pi n</math> for all integers <math>n</math>. Therefore, we want <math>\frac{1}{x} = \pi n</math> on <math>\frac{1}{10000} \le x \le \frac{1}{1000}</math>, so <math>1000 \le \frac 1x = \pi n \le 10000</math>. There are <math>\frac{10000-1000}{\pi} \approx \boxed{2900} \Rightarrow \mathrm{(A)}</math> solutions for <math>n</math> on this interval.  </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==Solution==</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">We know that <math>x</math> belongs to the interval <math>(0.0001,0.001)</math> for <math>sin(1/x)</math>. We see that when we plug in <math>x</math> into <math>sin(1/x)</math>, the argument <math>(1/x)</math> is always from the range <math>(1000, 10000)</math>. Therefore, the problem simply asks for all the zeros of <math>sin(x)</math> with <math>x</math> values between <math>(1000, 10000)</math>. We know that the <math>x</math> values of any sine graph is <math>\pi(n-1)</math> so, we see that values of <math>n</math> are any integer value from <math>320</math> to <math>3184</math> and therefore gives us an answer of approximately <math>2865</math> which is answer <math>\boxed{\text{(A)} 2900}</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">~Jske25</ins></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== See also ==</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== See also ==</div></td></tr>
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Jske25
https://artofproblemsolving.com/wiki/index.php?title=2003_AMC_12B_Problems/Problem_23&diff=53997&oldid=prev
Nathan wailes at 14:27, 4 July 2013
2013-07-04T14:27:25Z
<p></p>
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Nathan wailes
https://artofproblemsolving.com/wiki/index.php?title=2003_AMC_12B_Problems/Problem_23&diff=22922&oldid=prev
Azjps: s
2008-02-03T16:42:28Z
<p>s</p>
<p><b>New page</b></p><div>== Problem ==<br />
The number of <math>x</math>-intercepts on the graph of <math>y=\sin(1/x)</math> in the interval <math>(0.0001,0.001)</math> is closest to<br />
<br />
<math>\mathrm{(A)}\ 2900<br />
\qquad\mathrm{(B)}\ 3000<br />
\qquad\mathrm{(C)}\ 3100<br />
\qquad\mathrm{(D)}\ 3200<br />
\qquad\mathrm{(E)}\ 3300</math><br />
== Solution ==<br />
The function <math>f(x) = \sin x</math> has roots in the form of <math>\pi n</math> for all integers <math>n</math>. Therefore, we want <math>\frac{1}{x} = \pi n</math> on <math>\frac{1}{10000} \le x \le \frac{1}{1000}</math>, so <math>1000 \le \frac 1x = \pi n \le 10000</math>. There are <math>\frac{10000-1000}{\pi} \approx \boxed{2900} \Rightarrow \mathrm{(A)}</math> solutions for <math>n</math> on this interval. <br />
<br />
== See also ==<br />
{{AMC12 box|year=2003|ab=B|num-b=22|num-a=24}}<br />
<br />
[[Category:Introductory Trigonometry Problems]]</div>
Azjps