https://artofproblemsolving.com/wiki/index.php?title=1987_AHSME_Problems/Problem_19&feed=atom&action=history
1987 AHSME Problems/Problem 19 - Revision history
2024-03-28T16:35:40Z
Revision history for this page on the wiki
MediaWiki 1.31.1
https://artofproblemsolving.com/wiki/index.php?title=1987_AHSME_Problems/Problem_19&diff=96132&oldid=prev
Flyhawkeye: /* Solution */ Corrected solution and added actual approximate value
2018-07-12T12:02:07Z
<p><span dir="auto"><span class="autocomment">Solution: </span> Corrected solution and added actual approximate value</span></p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 12:02, 12 July 2018</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l10" >Line 10:</td>
<td colspan="2" class="diff-lineno">Line 10:</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>== 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 have <math>\sqrt{65} > 8 > 7.5</math>. Also <math>7.5^2 = (7 + 0.5)^2 = 7^2 + 2 \cdot 7 \cdot 0.5 + 0.5^2 = 49 + 7 + 0.25 = 56.25 < 63</math>, so <math>\sqrt{63} > 7.5</math>. Thus <math>\sqrt{65} + \sqrt{63} > 7.5 + 7.5 = 15</math>. Now notice that <math>\sqrt{65} - \sqrt{63} = \frac{(\sqrt{65} - \sqrt{63})(\sqrt{65} + \sqrt{63})}{\sqrt{65} + \sqrt{63}} = \frac{2}{\sqrt{65} + \sqrt{63}}</math>, so <math>\sqrt{65} - \sqrt{63} < \frac{2}{15} = 0.1333333...</math>, so the answer must be <math>A</math> or <math>B</math>. To determine which, we write <math>\sqrt{65} - \sqrt{63} > 0.125 \iff 65 - 2\sqrt{65 \cdot 63} + 63 > 0.015625 \iff 128 - 0.015625 > 2\sqrt{4095} \iff \sqrt{4095} < 64 - 0.0078125 \iff 4095 < 4096 - <del class="diffchange diffchange-inline">32 </del>\cdot 0.0078125 + 0.0078125^2 = 4096 - <del class="diffchange diffchange-inline">0.25 </del>+ 0.0078125^2</math> which is true. Hence as the expression is greater than <math>0.125</math>, and less than or equal to <math>0.13</math> (since we showed it is certainly less than <math>0.1333333...</math>), it is closest to <math>0.13</math>, which is answer <math>\boxed{\text{B}}</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 have <math>\sqrt{65} > 8 > 7.5</math>. Also <math>7.5^2 = (7 + 0.5)^2 = 7^2 + 2 \cdot 7 \cdot 0.5 + 0.5^2 = 49 + 7 + 0.25 = 56.25 < 63</math>, so <math>\sqrt{63} > 7.5</math>. Thus <math>\sqrt{65} + \sqrt{63} > 7.5 + 7.5 = 15</math>. Now notice that <math>\sqrt{65} - \sqrt{63} = \frac{(\sqrt{65} - \sqrt{63})(\sqrt{65} + \sqrt{63})}{\sqrt{65} + \sqrt{63}} = \frac{2}{\sqrt{65} + \sqrt{63}}</math>, so <math>\sqrt{65} - \sqrt{63} < \frac{2}{15} = 0.1333333...</math>, so the answer must be <math>A</math> or <math>B</math>. To determine which, we write <math>\sqrt{65} - \sqrt{63} > 0.125 \iff 65 - 2\sqrt{65 \cdot 63} + 63 > 0.015625 \iff 128 - 0.015625 > 2\sqrt{4095} \iff \sqrt{4095} < 64 - 0.0078125 \iff 4095 < 4096 - <ins class="diffchange diffchange-inline">128 </ins>\cdot 0.0078125 + 0.0078125^2 = 4096 - <ins class="diffchange diffchange-inline">1 </ins>+ 0.0078125^2</math> which is true. Hence as the expression is greater than <math>0.125</math>, and less than or equal to <math>0.13</math> (since we showed it is certainly less than <math>0.1333333...</math>), it is closest to <math>0.13</math>, which is answer <math>\boxed{\text{B}}</math>.</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> </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 class="diffchange diffchange-inline">(<math>\sqrt{65} - \sqrt{63}</math> is approximately equal to <math>0.125003815</math>)</ins></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>== 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>
</table>
Flyhawkeye
https://artofproblemsolving.com/wiki/index.php?title=1987_AHSME_Problems/Problem_19&diff=92558&oldid=prev
Hapaxoromenon: Fixed some formatting
2018-03-01T18:49:35Z
<p>Fixed some formatting</p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<col class="diff-marker" />
<col class="diff-content" />
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 18:49, 1 March 2018</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l10" >Line 10:</td>
<td colspan="2" class="diff-lineno">Line 10:</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>== 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 have <math>\sqrt{65} > 8 > 7.5</math>. Also <math>7.5^2 = (7 + 0.5)^2 = 7^2 + 2 \cdot 7 \cdot 0.5 + 0.5^2 = 49 + 7 + 0.25 = 56.25 < 63</math>, so <math>\sqrt{63} > 7.5</math>. Thus <math>\sqrt{65} + \sqrt{63} > 7.5 + 7.5 = 15</math>. Now notice that <math>\sqrt{65} - \sqrt{63} = \frac{(\sqrt{65} - \sqrt{63})(\sqrt{65} + \sqrt{63})}{\sqrt{65} + \sqrt{63}} = \frac{2}{\sqrt{65} + \sqrt{63}}</math>, so <math>\sqrt{65} - \sqrt{63} < \frac{2}{15} = 0.1333333...</math>, so the answer must be <math>A</math> or <math>B</math>. To determine which, we write <math>\sqrt{65} - \sqrt{63} > 0.125 \iff 65 - 2\sqrt{65 \cdot 63} + 63 > 0.015625 \iff 128 - 0.015625 > 2\sqrt{4095} \iff \sqrt{4095} < 64 - 0.0078125 \iff 4095 < 4096 - 32 \cdot 0.0078125 + 0.0078125^2 = 4096 - 0.25 + 0.0078125^2</math><del class="diffchange diffchange-inline">, </del>which is true. Hence as the expression is greater than <math>0.125</math>, and less than or equal to <math>0.13</math> (since we showed it is certainly less than <math>0.1333333...</math>), it is closest to <math>0.13</math>, which is answer <math>\boxed{\text{B}}</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 have <math>\sqrt{65} > 8 > 7.5</math>. Also <math>7.5^2 = (7 + 0.5)^2 = 7^2 + 2 \cdot 7 \cdot 0.5 + 0.5^2 = 49 + 7 + 0.25 = 56.25 < 63</math>, so <math>\sqrt{63} > 7.5</math>. Thus <math>\sqrt{65} + \sqrt{63} > 7.5 + 7.5 = 15</math>. Now notice that <math>\sqrt{65} - \sqrt{63} = \frac{(\sqrt{65} - \sqrt{63})(\sqrt{65} + \sqrt{63})}{\sqrt{65} + \sqrt{63}} = \frac{2}{\sqrt{65} + \sqrt{63}}</math>, so <math>\sqrt{65} - \sqrt{63} < \frac{2}{15} = 0.1333333...</math>, so the answer must be <math>A</math> or <math>B</math>. To determine which, we write <math>\sqrt{65} - \sqrt{63} > 0.125 \iff 65 - 2\sqrt{65 \cdot 63} + 63 > 0.015625 \iff 128 - 0.015625 > 2\sqrt{4095} \iff \sqrt{4095} < 64 - 0.0078125 \iff 4095 < 4096 - 32 \cdot 0.0078125 + 0.0078125^2 = 4096 - 0.25 + 0.0078125^2</math> which is true. Hence as the expression is greater than <math>0.125</math>, and less than or equal to <math>0.13</math> (since we showed it is certainly less than <math>0.1333333...</math>), it is closest to <math>0.13</math>, which is answer <math>\boxed{\text{B}}</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>== 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>
</table>
Hapaxoromenon
https://artofproblemsolving.com/wiki/index.php?title=1987_AHSME_Problems/Problem_19&diff=92557&oldid=prev
Hapaxoromenon: Fixed some more LaTeX
2018-03-01T18:49:00Z
<p>Fixed some more LaTeX</p>
<table class="diff diff-contentalign-left" data-mw="interface">
<col class="diff-marker" />
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<col class="diff-marker" />
<col class="diff-content" />
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 18:49, 1 March 2018</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l10" >Line 10:</td>
<td colspan="2" class="diff-lineno">Line 10:</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>== 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 have <math>\sqrt{65} > 8 > 7.5</math>. Also <math>7.5^2 = (7 + 0.5)^2 = 7^2 + 2 \cdot 7 \cdot 0.5 + 0.5^2 = 49 + 7 + 0.25 = 56.25 < 63</math>, so <math>\sqrt{63} > 7.5</math>. Thus <math>\sqrt{65} + \sqrt{63} > 7.5 + 7.5 = 15</math>. Now notice that <math>\sqrt{65} - \sqrt{63} = \frac{(\sqrt{65} - \sqrt{63})(\sqrt{65} + \sqrt{63})}{\sqrt{65} + \sqrt{63}} = \frac{2}{\sqrt{65} + \sqrt{63}</math>, so <math>\sqrt{65} - \sqrt{63} < \frac{2}{15} = 0.1333333...</math>, so the answer must be <math>A</math> or <math>B</math>. To determine which, we write <math>\sqrt{65} - \sqrt{63} > 0.125 \iff 65 - 2\sqrt{65 \cdot 63} + 63 > 0.015625 \iff 128 - 0.015625 > 2\sqrt{4095} \iff \sqrt{4095} < 64 - 0.0078125 \iff 4095 < 4096 - 32 \cdot 0.0078125 + 0.0078125^2 = 4096 - 0.25 + 0.0078125^2</math>, which is true. Hence as the expression is greater than <math>0.125</math>, and less than or equal to <math>0.13</math> (since we showed it is certainly less than <math>0.1333333...</math>), it is closest to <math>0.13</math>, which is answer <math>\boxed{\text{B}}</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 have <math>\sqrt{65} > 8 > 7.5</math>. Also <math>7.5^2 = (7 + 0.5)^2 = 7^2 + 2 \cdot 7 \cdot 0.5 + 0.5^2 = 49 + 7 + 0.25 = 56.25 < 63</math>, so <math>\sqrt{63} > 7.5</math>. Thus <math>\sqrt{65} + \sqrt{63} > 7.5 + 7.5 = 15</math>. Now notice that <math>\sqrt{65} - \sqrt{63} = \frac{(\sqrt{65} - \sqrt{63})(\sqrt{65} + \sqrt{63})}{\sqrt{65} + \sqrt{63}} = \frac{2}{\sqrt{65} + \sqrt{63<ins class="diffchange diffchange-inline">}</ins>}</math>, so <math>\sqrt{65} - \sqrt{63} < \frac{2}{15} = 0.1333333...</math>, so the answer must be <math>A</math> or <math>B</math>. To determine which, we write <math>\sqrt{65} - \sqrt{63} > 0.125 \iff 65 - 2\sqrt{65 \cdot 63} + 63 > 0.015625 \iff 128 - 0.015625 > 2\sqrt{4095} \iff \sqrt{4095} < 64 - 0.0078125 \iff 4095 < 4096 - 32 \cdot 0.0078125 + 0.0078125^2 = 4096 - 0.25 + 0.0078125^2</math>, which is true. Hence as the expression is greater than <math>0.125</math>, and less than or equal to <math>0.13</math> (since we showed it is certainly less than <math>0.1333333...</math>), it is closest to <math>0.13</math>, which is answer <math>\boxed{\text{B}}</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>== 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>
</table>
Hapaxoromenon
https://artofproblemsolving.com/wiki/index.php?title=1987_AHSME_Problems/Problem_19&diff=92556&oldid=prev
Hapaxoromenon: Fixed some LaTeX
2018-03-01T18:48:32Z
<p>Fixed some LaTeX</p>
<table class="diff diff-contentalign-left" data-mw="interface">
<col class="diff-marker" />
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<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 18:48, 1 March 2018</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l10" >Line 10:</td>
<td colspan="2" class="diff-lineno">Line 10:</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>== 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 have <math>\sqrt{65} > 8 > 7.5</math>. Also <math>7.5^2 = (7 + 0.5)^2 = 7^2 + 2 \cdot 7 \cdot 0.5 + 0.5^2 = 49 + 7 + 0.25 = 56.25 < 63</math>, so <math>\sqrt{63} > 7.5</math>. Thus <math>\sqrt{65} + \sqrt{63} > 7.5 + 7.5 = 15</math>. Now notice that <math>\sqrt{65} - \sqrt{63} = \frac{(\sqrt{65} - \sqrt{63})(\sqrt{65} + \sqrt{63})}{\sqrt{65} + \sqrt{63} = \frac{2}{\sqrt{65} + \sqrt{63}</math>, so <math>\sqrt{65} - \sqrt{63} < \frac{2}{15} = 0.1333333...</math>, so the answer must be <math>A</math> or <math>B</math>. To determine which, we write <math>\sqrt{65} - \sqrt{63} > 0.125 \iff 65 - 2\sqrt{65 \cdot 63} + 63 > 0.015625 \iff 128 - 0.015625 > 2\sqrt{4095} \iff \sqrt{4095} < 64 - 0.0078125 \iff 4095 < 4096 - 32 \cdot 0.0078125 + 0.0078125^2 = 4096 - 0.25 + 0.0078125^2</math>, which is true. Hence as the expression is greater than <math>0.125</math>, and less than or equal to <math>0.13</math> (since we showed it is certainly less than <math>0.1333333...</math>), it is closest to <math>0.13</math>, which is answer <math>\boxed{\text{B}}</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 have <math>\sqrt{65} > 8 > 7.5</math>. Also <math>7.5^2 = (7 + 0.5)^2 = 7^2 + 2 \cdot 7 \cdot 0.5 + 0.5^2 = 49 + 7 + 0.25 = 56.25 < 63</math>, so <math>\sqrt{63} > 7.5</math>. Thus <math>\sqrt{65} + \sqrt{63} > 7.5 + 7.5 = 15</math>. Now notice that <math>\sqrt{65} - \sqrt{63} = \frac{(\sqrt{65} - \sqrt{63})(\sqrt{65} + \sqrt{63})}{\sqrt{65} + \sqrt{63<ins class="diffchange diffchange-inline">}</ins>} = \frac{2}{\sqrt{65} + \sqrt{63}</math>, so <math>\sqrt{65} - \sqrt{63} < \frac{2}{15} = 0.1333333...</math>, so the answer must be <math>A</math> or <math>B</math>. To determine which, we write <math>\sqrt{65} - \sqrt{63} > 0.125 \iff 65 - 2\sqrt{65 \cdot 63} + 63 > 0.015625 \iff 128 - 0.015625 > 2\sqrt{4095} \iff \sqrt{4095} < 64 - 0.0078125 \iff 4095 < 4096 - 32 \cdot 0.0078125 + 0.0078125^2 = 4096 - 0.25 + 0.0078125^2</math>, which is true. Hence as the expression is greater than <math>0.125</math>, and less than or equal to <math>0.13</math> (since we showed it is certainly less than <math>0.1333333...</math>), it is closest to <math>0.13</math>, which is answer <math>\boxed{\text{B}}</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>== 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>
</table>
Hapaxoromenon
https://artofproblemsolving.com/wiki/index.php?title=1987_AHSME_Problems/Problem_19&diff=92555&oldid=prev
Hapaxoromenon: Added a solution with explanation
2018-03-01T18:47:54Z
<p>Added a solution with explanation</p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 18:47, 1 March 2018</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l8" >Line 8:</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>\textbf{(D)}\ .15 \qquad</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>\textbf{(D)}\ .15 \qquad</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>\textbf{(E)}\ .16  </math>   </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>\textbf{(E)}\ .16  </math>   </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 have <math>\sqrt{65} > 8 > 7.5</math>. Also <math>7.5^2 = (7 + 0.5)^2 = 7^2 + 2 \cdot 7 \cdot 0.5 + 0.5^2 = 49 + 7 + 0.25 = 56.25 < 63</math>, so <math>\sqrt{63} > 7.5</math>. Thus <math>\sqrt{65} + \sqrt{63} > 7.5 + 7.5 = 15</math>. Now notice that <math>\sqrt{65} - \sqrt{63} = \frac{(\sqrt{65} - \sqrt{63})(\sqrt{65} + \sqrt{63})}{\sqrt{65} + \sqrt{63} = \frac{2}{\sqrt{65} + \sqrt{63}</math>, so <math>\sqrt{65} - \sqrt{63} < \frac{2}{15} = 0.1333333...</math>, so the answer must be <math>A</math> or <math>B</math>. To determine which, we write <math>\sqrt{65} - \sqrt{63} > 0.125 \iff 65 - 2\sqrt{65 \cdot 63} + 63 > 0.015625 \iff 128 - 0.015625 > 2\sqrt{4095} \iff \sqrt{4095} < 64 - 0.0078125 \iff 4095 < 4096 - 32 \cdot 0.0078125 + 0.0078125^2 = 4096 - 0.25 + 0.0078125^2</math>, which is true. Hence as the expression is greater than <math>0.125</math>, and less than or equal to <math>0.13</math> (since we showed it is certainly less than <math>0.1333333...</math>), it is closest to <math>0.13</math>, which is answer <math>\boxed{\text{B}}</math>.</ins></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>== 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>
</table>
Hapaxoromenon
https://artofproblemsolving.com/wiki/index.php?title=1987_AHSME_Problems/Problem_19&diff=65690&oldid=prev
Timneh: Created page with "==Problem== Which of the following is closest to <math>\sqrt{65}-\sqrt{63}</math>? <math>\textbf{(A)}\ .12 \qquad \textbf{(B)}\ .13 \qquad \textbf{(C)}\ .14 \qquad \textbf{(D)}..."
2014-10-23T12:48:17Z
<p>Created page with "==Problem== Which of the following is closest to <math>\sqrt{65}-\sqrt{63}</math>? <math>\textbf{(A)}\ .12 \qquad \textbf{(B)}\ .13 \qquad \textbf{(C)}\ .14 \qquad \textbf{(D)}..."</p>
<p><b>New page</b></p><div>==Problem==<br />
<br />
Which of the following is closest to <math>\sqrt{65}-\sqrt{63}</math>?<br />
<br />
<math>\textbf{(A)}\ .12 \qquad<br />
\textbf{(B)}\ .13 \qquad<br />
\textbf{(C)}\ .14 \qquad<br />
\textbf{(D)}\ .15 \qquad<br />
\textbf{(E)}\ .16 </math> <br />
<br />
== See also ==<br />
{{AHSME box|year=1987|num-b=18|num-a=20}} <br />
<br />
[[Category: Intermediate Algebra Problems]]<br />
{{MAA Notice}}</div>
Timneh