{"id":2377,"date":"2021-11-08T23:06:00","date_gmt":"2021-11-09T03:06:00","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=2377"},"modified":"2024-10-25T11:27:47","modified_gmt":"2024-10-25T15:27:47","slug":"storing-cans-on-shelf","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=2377","title":{"rendered":"Storing Cans on Shelf"},"content":{"rendered":"\n<p>If cans can be placed on top of the one another straight up, how many cylindrical cans $4$ inches in diameter and $6$ inches high can be stored on a shelf $2$ feet wide and $6$ feet long if the shelf is $1$ feet down from the ceiling? Courtney<em> CML Questions Grade 4-6: Problem 362<\/em> <\/p>\n\n\n\n<p>The standard answer given by CML is $$\\dfrac{(2\\times 12)\\times (6\\times 12)}{4\\times 4}\\times\\dfrac{1\\times 12}{6}=216$$ That is to store $108$ cans at the bottom, and another $108$ cans on the top. <\/p>\n\n\n\n<p>However, if the cans are stored in the following pattern on the shelf, there will be $$(6\\times 10+5\\times 10)\\times 2=220$$ with $2$ more cans stored on the shelf:<\/p>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" src=\"https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2022\/01\/Screen-Shot-2022-01-01-at-5.27.24-PM-1024x372.png\" alt=\"\" class=\"wp-image-2381\" width=\"512\" height=\"186\" srcset=\"https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2022\/01\/Screen-Shot-2022-01-01-at-5.27.24-PM-1024x372.png 1024w, https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2022\/01\/Screen-Shot-2022-01-01-at-5.27.24-PM-300x109.png 300w, https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2022\/01\/Screen-Shot-2022-01-01-at-5.27.24-PM-768x279.png 768w, https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2022\/01\/Screen-Shot-2022-01-01-at-5.27.24-PM-1536x558.png 1536w, https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2022\/01\/Screen-Shot-2022-01-01-at-5.27.24-PM-2048x743.png 2048w\" sizes=\"(max-width: 512px) 100vw, 512px\" \/><\/figure>\n\n\n\n<p>The above packing is more space efficient, as the required length $L$ for placing $n$ columns of cans with diameter as $d$ is $$L=d+(n-1)\\cdot\\dfrac{\\sqrt{3}}{2}\\cdot d$$ Given the shelf length as $l$, we have $$n=1+\\Big\\lfloor \\dfrac{2\\cdot\\sqrt{3}\\cdot(l-d)}{3\\cdot d}\\Big\\rfloor$$<\/p>\n\n\n\n<p>If $l=72$ and $d=4$, then $n$=20, with $$L=4+38\\sqrt{3}\\approx69.816\\lt 72$$ leaving more than 2 inches to spare.<\/p>\n\n\n\n<p>If $l=32$ or $l=60$, can you verify that more tighter packing can be done?<\/p>\n","protected":false},"excerpt":{"rendered":"<p>If cans can be placed on top of the one another straight up, how many cylindrical cans $4$ inches in diameter and $6$ inches high can be stored on a shelf $2$ feet wide and $6$ feet long if the &hellip; <a href=\"https:\/\/mathfun4kids.com\/mlog\/?p=2377\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[13,9],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/2377"}],"collection":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2377"}],"version-history":[{"count":20,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/2377\/revisions"}],"predecessor-version":[{"id":2398,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/2377\/revisions\/2398"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2377"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2377"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2377"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}