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2014 AMC 12A Problems/Problem 17 - Revision history
2024-03-28T22:24:39Z
Revision history for this page on the wiki
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Sl hc: /* Solution 1 */
2022-06-13T06:42:10Z
<p><span dir="auto"><span class="autocomment">Solution 1</span></span></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 06:42, 13 June 2022</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l34" >Line 34:</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>Let <math>O</math> be the center of any of the top spheres (you choose!). We have <math>AO=1\cdot\sqrt{2}</math>, and <math>CO=3</math>, so <math>AC=\sqrt{3^2-\left(\sqrt2\right)^2}=\sqrt{7}</math>. Similarly, <math>BC=\sqrt{7}</math>. <math>\overline{AD}</math> and <math>\overline{BE}</math> are clearly equal to the radius of the small spheres, <math>1</math>. Thus the total height is <math>AD+AC+BC+BE=2+2\sqrt7</math>, or <math>\boxed{\textbf{(A)}}</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>Let <math>O</math> be the center of any of the top spheres (you choose!). We have <math>AO=1\cdot\sqrt{2}</math>, and <math>CO=3</math>, so <math>AC=\sqrt{3^2-\left(\sqrt2\right)^2}=\sqrt{7}</math>. Similarly, <math>BC=\sqrt{7}</math>. <math>\overline{AD}</math> and <math>\overline{BE}</math> are clearly equal to the radius of the small spheres, <math>1</math>. Thus the total height is <math>AD+AC+BC+BE=2+2\sqrt7</math>, or <math>\boxed{\textbf{(A)}}</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="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><del class="diffchange diffchange-inline">(Solution by </del>AwesomeToad<del class="diffchange diffchange-inline">)</del></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><ins class="diffchange diffchange-inline">~</ins>AwesomeToad</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>==Solution 2==</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 2==</div></td></tr>
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Sl hc
https://artofproblemsolving.com/wiki/index.php?title=2014_AMC_12A_Problems/Problem_17&diff=174782&oldid=prev
Sl hc: /* Solution 2 */
2022-06-13T06:41:44Z
<p><span dir="auto"><span class="autocomment">Solution 2</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 06:41, 13 June 2022</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l44" >Line 44:</td>
<td colspan="2" class="diff-lineno">Line 44:</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><math>AC</math> is the radii of the spheres, so <math>AC=2+1=3</math>. <math>CE</math> is the shortest length between the center of a small sphere and the edge of the prism, so <math>CE=\sqrt{2}</math>. Similarly, <math>AF=2\sqrt{2}</math>. Since <math>CEFB</math> is a rectangle, <math>BF=CE=\sqrt{2}</math>. Since <math>AF=2\sqrt{2}</math>, <math>AB=AF - BF = \sqrt{2}</math>. Then, <math>BC=\sqrt{3^2-\left(\sqrt2\right)^2}=\sqrt{7}=EF</math>. <math>DE</math> is the length from <math>C</math> to the top of the prism or <math>1</math>. Thus, <math>DF=DE+EF=1+\sqrt{7}</math>. The prism is symmetrical, so <math>h=2DF=\boxed{\textbf{(A)}}</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><math>AC</math> is the radii of the spheres, so <math>AC=2+1=3</math>. <math>CE</math> is the shortest length between the center of a small sphere and the edge of the prism, so <math>CE=\sqrt{2}</math>. Similarly, <math>AF=2\sqrt{2}</math>. Since <math>CEFB</math> is a rectangle, <math>BF=CE=\sqrt{2}</math>. Since <math>AF=2\sqrt{2}</math>, <math>AB=AF - BF = \sqrt{2}</math>. Then, <math>BC=\sqrt{3^2-\left(\sqrt2\right)^2}=\sqrt{7}=EF</math>. <math>DE</math> is the length from <math>C</math> to the top of the prism or <math>1</math>. Thus, <math>DF=DE+EF=1+\sqrt{7}</math>. The prism is symmetrical, so <math>h=2DF=\boxed{\textbf{(A)}}</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="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><del class="diffchange diffchange-inline">(Solution by </del>BJHHar<del class="diffchange diffchange-inline">)</del></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><ins class="diffchange diffchange-inline">~</ins>BJHHar</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>==Solution 3==</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 3==</div></td></tr>
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Sl hc
https://artofproblemsolving.com/wiki/index.php?title=2014_AMC_12A_Problems/Problem_17&diff=174781&oldid=prev
Sl hc: Adding box to answer choice
2022-06-13T06:41:05Z
<p>Adding box to answer choice</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 06:41, 13 June 2022</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l48" >Line 48:</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 3==</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 3==</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="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>Take a cross section and see that h is made up of two radii of the <del class="diffchange diffchange-inline">circle </del>plus some radical expression. The only choice satisfying this condition is (<del class="diffchange diffchange-inline">a</del>)</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>Take a cross section and <ins class="diffchange diffchange-inline">you will </ins>see that h is made up of <ins class="diffchange diffchange-inline">the </ins>two radii of the <ins class="diffchange diffchange-inline">circles </ins>plus some radical expression. The only choice satisfying this condition is <ins class="diffchange diffchange-inline"><math>\boxed{\textbf{</ins>(<ins class="diffchange diffchange-inline">A</ins>)<ins class="diffchange diffchange-inline">}}</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>
<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>{{AMC12 box|year=2014|ab=A|num-b=16|num-a=18}}</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>{{AMC12 box|year=2014|ab=A|num-b=16|num-a=18}}</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>{{MAA Notice}}</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>{{MAA Notice}}</div></td></tr>
</table>
Sl hc
https://artofproblemsolving.com/wiki/index.php?title=2014_AMC_12A_Problems/Problem_17&diff=174780&oldid=prev
Sl hc: /* Solution 3 */
2022-06-13T06:39:36Z
<p><span dir="auto"><span class="autocomment">Solution 3</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 06:39, 13 June 2022</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l48" >Line 48:</td>
<td colspan="2" class="diff-lineno">Line 48:</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 3==</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 3==</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="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><del class="diffchange diffchange-inline">take </del>a cross section and see that h is made up of two radii of the circle plus some radical expression. <del class="diffchange diffchange-inline">the </del>only choice satisfying this condition is (a)</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><ins class="diffchange diffchange-inline">Take </ins>a cross section and see that h is made up of two radii of the circle plus some radical expression. <ins class="diffchange diffchange-inline">The </ins>only choice satisfying this condition is (a)</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>
<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>{{AMC12 box|year=2014|ab=A|num-b=16|num-a=18}}</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>{{AMC12 box|year=2014|ab=A|num-b=16|num-a=18}}</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>{{MAA Notice}}</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>{{MAA Notice}}</div></td></tr>
</table>
Sl hc
https://artofproblemsolving.com/wiki/index.php?title=2014_AMC_12A_Problems/Problem_17&diff=159559&oldid=prev
Sosiaops: made formatting clearer
2021-08-04T17:11:09Z
<p>made formatting clearer</p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 17:11, 4 August 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;"></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;"></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="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><math>AC</math> is the radii of the spheres, so <math>AC=2+1=3</math>. <math>CE</math> is the shortest length between the center of a small sphere and the edge of the prism, so <math>CE=\sqrt{2}</math>. Similarly, <math>AF=2\sqrt{2}</math>. Since <math>CEFB</math> is a rectangle, <math>BF=CE=\sqrt{2}</math>. Since <math>AF=2\sqrt{2}</math>, <math>AB=AF - BF = \sqrt{2}</math>. Then, <math>BC=\sqrt{3^2-\sqrt2^2}=\sqrt{7}=EF</math>. <math>DE</math> is the length from <math>C</math> to the top of the prism or <math>1</math>. Thus, <math>DF=DE+EF=1+\sqrt{7}</math>. The prism is symmetrical, so <math>h=2DF=\boxed{\textbf{(A)}}</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><math>AC</math> is the radii of the spheres, so <math>AC=2+1=3</math>. <math>CE</math> is the shortest length between the center of a small sphere and the edge of the prism, so <math>CE=\sqrt{2}</math>. Similarly, <math>AF=2\sqrt{2}</math>. Since <math>CEFB</math> is a rectangle, <math>BF=CE=\sqrt{2}</math>. Since <math>AF=2\sqrt{2}</math>, <math>AB=AF - BF = \sqrt{2}</math>. Then, <math>BC=\sqrt{3^2-<ins class="diffchange diffchange-inline">\left(</ins>\sqrt2<ins class="diffchange diffchange-inline">\right)</ins>^2}=\sqrt{7}=EF</math>. <math>DE</math> is the length from <math>C</math> to the top of the prism or <math>1</math>. Thus, <math>DF=DE+EF=1+\sqrt{7}</math>. The prism is symmetrical, so <math>h=2DF=\boxed{\textbf{(A)}}</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>(Solution by BJHHar)</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 by BJHHar)</div></td></tr>
</table>
Sosiaops
https://artofproblemsolving.com/wiki/index.php?title=2014_AMC_12A_Problems/Problem_17&diff=159558&oldid=prev
Sosiaops: made formatting clearer
2021-08-04T17:10:16Z
<p>made formatting clearer</p>
<table class="diff diff-contentalign-left" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<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 17:10, 4 August 2021</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l32" >Line 32:</td>
<td colspan="2" class="diff-lineno">Line 32:</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>Let <math>A</math> be the point in the same plane as the centers of the top spheres equidistant from said centers. Let <math>B</math> be the analogous point for the bottom spheres, and let <math>C</math> be the midpoint of <math>\overline{AB}</math> and the center of the large sphere. Let <math>D</math> and <math>E</math> be the points at which line <math>AB</math> intersects the top of the box and the bottom, respectively.</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>Let <math>A</math> be the point in the same plane as the centers of the top spheres equidistant from said centers. Let <math>B</math> be the analogous point for the bottom spheres, and let <math>C</math> be the midpoint of <math>\overline{AB}</math> and the center of the large sphere. Let <math>D</math> and <math>E</math> be the points at which line <math>AB</math> intersects the top of the box and the bottom, respectively.</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="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>Let <math>O</math> be the center of any of the top spheres (you choose!). We have <math>AO=1\cdot\sqrt{2}</math>, and <math>CO=3</math>, so <math>AC=\sqrt{3^2-\sqrt2^2}=\sqrt{7}</math>. Similarly, <math>BC=\sqrt{7}</math>. <math>\overline{AD}</math> and <math>\overline{BE}</math> are clearly equal to the radius of the small spheres, <math>1</math>. Thus the total height is <math>AD+AC+BC+BE=2+2\sqrt7</math>, or <math>\boxed{\textbf{(A)}}</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>Let <math>O</math> be the center of any of the top spheres (you choose!). We have <math>AO=1\cdot\sqrt{2}</math>, and <math>CO=3</math>, so <math>AC=\sqrt{3^2-<ins class="diffchange diffchange-inline">\left(</ins>\sqrt2<ins class="diffchange diffchange-inline">\right)</ins>^2}=\sqrt{7}</math>. Similarly, <math>BC=\sqrt{7}</math>. <math>\overline{AD}</math> and <math>\overline{BE}</math> are clearly equal to the radius of the small spheres, <math>1</math>. Thus the total height is <math>AD+AC+BC+BE=2+2\sqrt7</math>, or <math>\boxed{\textbf{(A)}}</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>(Solution by AwesomeToad)</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 by AwesomeToad)</div></td></tr>
</table>
Sosiaops
https://artofproblemsolving.com/wiki/index.php?title=2014_AMC_12A_Problems/Problem_17&diff=127665&oldid=prev
Summer279: /* Solution */
2020-07-07T13:08:53Z
<p><span dir="auto"><span class="autocomment">Solution</span></span></p>
<table class="diff diff-contentalign-left" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<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 13:08, 7 July 2020</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l28" >Line 28:</td>
<td colspan="2" class="diff-lineno">Line 28:</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>[[Category: Introductory Geometry Problems]]</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>[[Category: Introductory Geometry Problems]]</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="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>==Solution==</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>==Solution <ins class="diffchange diffchange-inline">1</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>Let <math>A</math> be the point in the same plane as the centers of the top spheres equidistant from said centers. Let <math>B</math> be the analogous point for the bottom spheres, and let <math>C</math> be the midpoint of <math>\overline{AB}</math> and the center of the large sphere. Let <math>D</math> and <math>E</math> be the points at which line <math>AB</math> intersects the top of the box and the bottom, respectively.</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>Let <math>A</math> be the point in the same plane as the centers of the top spheres equidistant from said centers. Let <math>B</math> be the analogous point for the bottom spheres, and let <math>C</math> be the midpoint of <math>\overline{AB}</math> and the center of the large sphere. Let <math>D</math> and <math>E</math> be the points at which line <math>AB</math> intersects the top of the box and the bottom, respectively.</div></td></tr>
</table>
Summer279
https://artofproblemsolving.com/wiki/index.php?title=2014_AMC_12A_Problems/Problem_17&diff=120843&oldid=prev
S.das93: /* Solution 3 */
2020-04-12T00:42:16Z
<p><span dir="auto"><span class="autocomment">Solution 3</span></span></p>
<table class="diff diff-contentalign-left" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<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 00:42, 12 April 2020</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l48" >Line 48:</td>
<td colspan="2" class="diff-lineno">Line 48:</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 3==</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 3==</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="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><del class="diffchange diffchange-inline">Use </del>the <del class="diffchange diffchange-inline">4 bottom spheres</del>. (<del class="diffchange diffchange-inline">In progress</del>)</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><ins class="diffchange diffchange-inline">take a cross section and see that h is made up of two radii of </ins>the <ins class="diffchange diffchange-inline">circle plus some radical expression</ins>. <ins class="diffchange diffchange-inline">the only choice satisfying this condition is </ins>(<ins class="diffchange diffchange-inline">a</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>
<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>{{AMC12 box|year=2014|ab=A|num-b=16|num-a=18}}</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>{{AMC12 box|year=2014|ab=A|num-b=16|num-a=18}}</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>{{MAA Notice}}</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>{{MAA Notice}}</div></td></tr>
</table>
S.das93
https://artofproblemsolving.com/wiki/index.php?title=2014_AMC_12A_Problems/Problem_17&diff=115771&oldid=prev
Epicjojo: /* Solution 2 */
2020-01-28T23:20:58Z
<p><span dir="auto"><span class="autocomment">Solution 2</span></span></p>
<table class="diff diff-contentalign-left" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<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 23:20, 28 January 2020</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l42" >Line 42:</td>
<td colspan="2" class="diff-lineno">Line 42:</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;"></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="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><math>AC</math> is the radii of the spheres, so <math>AC=2+1=3</math>. <math>CE</math> is the shortest length between the center of a small sphere and the edge of the prism, so <math>CE=\sqrt{2}</math>. Similarly, <math>AF=2\sqrt{2}</math>. Since <math>CEFB</math> is a rectangle, <math>CE=<del class="diffchange diffchange-inline">BF=2</del>\sqrt{2}</math>. Since <math>AF=2\sqrt{2}</math>, <math>AB=\sqrt{2}</math>. Then, <math>BC=\sqrt{3^2-\sqrt2^2}=\sqrt{7}=EF</math>. <math>DE</math> is the length from <math>C</math> to the top of the prism or <math>1</math>. Thus, <math>DF=DE+EF=1+\sqrt{7}</math>. The prism is symmetrical, so <math>h=2DF=\boxed{\textbf{(A)}}</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><math>AC</math> is the radii of the spheres, so <math>AC=2+1=3</math>. <math>CE</math> is the shortest length between the center of a small sphere and the edge of the prism, so <math>CE=\sqrt{2}</math>. Similarly, <math>AF=2\sqrt{2}</math>. Since <math>CEFB</math> is a rectangle, <math><ins class="diffchange diffchange-inline">BF=</ins>CE=\sqrt{2}</math>. Since <math>AF=2\sqrt{2}</math>, <math>AB<ins class="diffchange diffchange-inline">=AF - BF </ins>= \sqrt{2}</math>. Then, <math>BC=\sqrt{3^2-\sqrt2^2}=\sqrt{7}=EF</math>. <math>DE</math> is the length from <math>C</math> to the top of the prism or <math>1</math>. Thus, <math>DF=DE+EF=1+\sqrt{7}</math>. The prism is symmetrical, so <math>h=2DF=\boxed{\textbf{(A)}}</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>(Solution by BJHHar)</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 by BJHHar)</div></td></tr>
</table>
Epicjojo
https://artofproblemsolving.com/wiki/index.php?title=2014_AMC_12A_Problems/Problem_17&diff=109423&oldid=prev
Bjhhar: /* Solution 2 */
2019-08-27T02:41:06Z
<p><span dir="auto"><span class="autocomment">Solution 2</span></span></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 02:41, 27 August 2019</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l39" >Line 39:</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;"></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>[[File:2014 AMC 12A Problem 17.JPG|none|500px|caption]]</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>[[File:2014 AMC 12A Problem 17.JPG|none|500px|caption]]</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>Let <math>A</math> be the center of the sphere <del class="diffchange diffchange-inline">of radius <math>2</math> </del>and <math>C</math> be the center of any <del class="diffchange diffchange-inline">of the spheres</del>. Let <math>D</math> be a vertex of the rectangular prism closest to point <math>C</math>. Let <math>F</math> be the point on the edge of the prism such that <math>\overline{DF}</math> and <math>\overline{AF}</math> are perpendicular. Let points <math>B</math> and point <math>E</math> lie on <math>\overline{AF}</math> and <math>\overline{DF}</math> respectively such that <math>\overline{CE}</math> and <math>\overline{CB}</math> are perpendicular at <math>C</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>Let <math>A</math> be the center of the <ins class="diffchange diffchange-inline">large </ins>sphere and <math>C</math> be the center of any <ins class="diffchange diffchange-inline">small sphere</ins>. Let <math>D</math> be a vertex of the rectangular prism closest to point <math>C</math>. Let <math>F</math> be the point on the edge of the prism such that <math>\overline{DF}</math> and <math>\overline{AF}</math> are perpendicular. Let points <math>B</math> and point <math>E</math> lie on <math>\overline{AF}</math> and <math>\overline{DF}</math> respectively such that <math>\overline{CE}</math> and <math>\overline{CB}</math> are perpendicular at <math>C</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="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><math>AC</math> is the radii of the spheres, so <math>AC=2+1=3</math>. <math>CE</math> is the shortest length between the center of <del class="diffchange diffchange-inline">the </del>small sphere and the edge of the prism, so <math>CE=\sqrt{2}</math>. Similarly, <math>AF=2\sqrt{2}</math>. Since <math>CEFB</math> is a rectangle, <math>CE=BF=2\sqrt{2}</math>. Since <math>AF=2\sqrt{2}</math>, <math>AB=\sqrt{2}</math>. Then, <math>BC=\sqrt{3^2-\sqrt2^2}=\sqrt{7}=EF</math>. <math>DE</math> is the length from <math>C</math> to the top of the prism or <math>1</math>. Thus, <math>DF=DE+EF=1+\sqrt{7}</math>. <del class="diffchange diffchange-inline">Since the </del>prism is symmetrical, <math>h=2DF=\boxed{\textbf{(A)}}</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> </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><math>AC</math> is the radii of the spheres, so <math>AC=2+1=3</math>. <math>CE</math> is the shortest length between the center of <ins class="diffchange diffchange-inline">a </ins>small sphere and the edge of the prism, so <math>CE=\sqrt{2}</math>. Similarly, <math>AF=2\sqrt{2}</math>. Since <math>CEFB</math> is a rectangle, <math>CE=BF=2\sqrt{2}</math>. Since <math>AF=2\sqrt{2}</math>, <math>AB=\sqrt{2}</math>. Then, <math>BC=\sqrt{3^2-\sqrt2^2}=\sqrt{7}=EF</math>. <math>DE</math> is the length from <math>C</math> to the top of the prism or <math>1</math>. Thus, <math>DF=DE+EF=1+\sqrt{7}</math>. <ins class="diffchange diffchange-inline">The </ins>prism is symmetrical, <ins class="diffchange diffchange-inline">so </ins><math>h=2DF=\boxed{\textbf{(A)}}</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>(Solution by BJHHar)</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 by BJHHar)</div></td></tr>
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Bjhhar