Problem 52985. Easy Sequences 47: Boxes with Prime Edges

Created by Ramon Villamangca

Solve Later

{"parts":[{"partUri":"/matlab/document.xml","contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?><w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>This is related to problem \"</w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/52911-easy-sequences-41-boxes-with-integer-edges\"><w:r><w:t>Easy Sequences 41: Boxes with Integer Edges</w:t></w:r></w:hyperlink><w:r><w:t>\". However, here we will be investigating a smaller-sized set, namely, the set of rectangular-faced boxes with edges representable by (not necessarily distinct) prime numbers.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Given a volume limit, </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>v</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t>, we are asked to </w:t></w:r><w:r><w:rPr><w:b/><w:u/></w:rPr><w:t>find the average volume</w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t> of all boxes with prime edges, whose volumes are less than or equal to </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>v</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t>.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>For example for </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>v=50</w:t></w:r></w:customXml><w:r><w:t> cubic units, all </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>11</w:t></w:r></w:customXml><w:r><w:t> possible prime-edged boxes are given below:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>            </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>\\{\\ (2,2,2),\\ (2,2,3),\\ (2,2,5),\\ (2,2,7),\\ (2,2,11),\\ \n(2,3,3),\\ (2,3,5),\\ (2,3,7),\\ (2,5,5),\\ (3,3,3),\\ (3,3,5)\\ \\}</w:t></w:r></w:customXml><w:r><w:t>.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Their average volume is: </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>(8+12+20+28+44+18+30+42+50+27+45)\\ /\\ 11 = 29.4545</w:t></w:r></w:customXml><w:r><w:t> cubic units.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Please present your answer in </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>4</w:t></w:r></w:customXml><w:r><w:t> decimal places.</w:t></w:r></w:p></w:body></w:document>","relationship":null}],"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","target":"/matlab/document.xml","relationshipId":"rId1"}]}

<div style = "text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; "><div style="block-size: 234px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 117px; transform-origin: 407px 117px; vertical-align: baseline; "><div style="block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 84.5px 8px; transform-origin: 84.5px 8px; unicode-bidi: normal; "><span style="">This is related to problem "</span></span><a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/52911-easy-sequences-41-boxes-with-integer-edges"><span style=""><span style="">Easy Sequences 41: Boxes with Integer Edges</span></span></a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 132.5px 8px; transform-origin: 132.5px 8px; unicode-bidi: normal; "><span style="">". However, here we will be investigating a smaller-sized set, namely, the set of rectangular-faced boxes with edges representable by (not necessarily distinct) prime numbers.</span></span></div><div style="block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 75.5px 8px; transform-origin: 75.5px 8px; unicode-bidi: normal; "><span style="font-weight: 700; ">Given a volume limit, </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">v</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 61.5px 8px; transform-origin: 61.5px 8px; unicode-bidi: normal; "><span style="font-weight: 700; ">, we are asked to </span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 86px 8px; transform-origin: 86px 8px; unicode-bidi: normal; "><span style="font-weight: 700; text-decoration: underline; text-decoration-line: underline; ">find the average volume</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 134px 8px; transform-origin: 134px 8px; unicode-bidi: normal; "><span style="font-weight: 700; "> of all boxes with prime edges, whose volumes are less than or equal to </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">v</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; "><span style="font-weight: 700; ">.</span></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 51.5px 8px; transform-origin: 51.5px 8px; unicode-bidi: normal; "><span style="">For example for </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" style="width: 43.5px; height: 18px;" width="43.5" height="18"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 48.5px 8px; transform-origin: 48.5px 8px; unicode-bidi: normal; "><span style=""> cubic units, all </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACQAAAAkCAYAAADhAJiYAAAAdklEQVRYhe2OwQmAMBAEp4d0kAbsw0bsIB3YQmqxB1uwF/0YCHiPyGEU3IHjXrMMCCG+RQTCC645loEdGDq6F0I1Vq511OOajEA6/3pz1OM2kRyjHldBClKQghSkIAX9MigCWzWaOrnm2AwsxmVgesgVQghRcwBVnXRb64UZ7QAAAABJRU5ErkJggg==" style="width: 18px; height: 18px;" width="18" height="18"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 144px 8px; transform-origin: 144px 8px; unicode-bidi: normal; "><span style=""> possible prime-edged boxes are given below:</span></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 24px 8px; transform-origin: 24px 8px; unicode-bidi: normal; "><span style="">            </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" style="width: 660.5px; height: 18.5px;" width="660.5" height="18.5"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; "><span style="">.</span></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 80px 8px; transform-origin: 80px 8px; unicode-bidi: normal; "><span style="">Their average volume is: </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" style="width: 445.5px; height: 18.5px;" width="445.5" height="18.5"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 37.5px 8px; transform-origin: 37.5px 8px; unicode-bidi: normal; "><span style=""> cubic units.</span></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 97.5px 8px; transform-origin: 97.5px 8px; unicode-bidi: normal; "><span style="">Please present your answer in </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: normal; font-weight: 400; color: rgb(0, 0, 0);">4</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 50.5px 8px; transform-origin: 50.5px 8px; unicode-bidi: normal; "><span style=""> decimal places.</span></span></div></div></div>

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Problem 52985. Easy Sequences 47: Boxes with Prime Edges

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Problem 52985. Easy Sequences 47: Boxes with Prime Edges

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