{"id":187005,"date":"2025-01-31T07:59:09","date_gmt":"2025-01-31T07:59:09","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=187005"},"modified":"2025-01-31T07:59:11","modified_gmt":"2025-01-31T07:59:11","slug":"a-soda-can-is-to-hold-12-fluid-ounces-suppose-that-the-bottom-and-top-are-twice-as-thick-as-the-sides","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/01\/31\/a-soda-can-is-to-hold-12-fluid-ounces-suppose-that-the-bottom-and-top-are-twice-as-thick-as-the-sides\/","title":{"rendered":"A soda can is to hold 12 fluid ounces. Suppose that the bottom and top are twice as thick as the sides"},"content":{"rendered":"\n<p>A soda can is to hold 12 fluid ounces. Suppose that the bottom and top are twice as thick as the sides. Find the dimensions of the can that minimize the amount of material used. (Hint: Instead of minimizing surface area, minimize the cost, which is proportional to the product of the thickness and the area.)<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Answer:<\/h3>\n\n\n\n<p>The optimal dimensions that minimize the material cost for a soda can holding 12 fluid ounces (\u2248 21.66 cubic inches) are:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Radius (r):<\/strong> ( \\sqrt[3]{\\frac{21.66}{2\\pi}} \\approx 1.54 ) inches<\/li>\n\n\n\n<li><strong>Height (h):<\/strong> ( \\frac{21.66}{\\pi(1.54)^2} \\approx 2.91 ) inches<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation:<\/h3>\n\n\n\n<p>To minimize material cost, we must consider both <strong>surface area<\/strong> and <strong>thickness distribution<\/strong>. The thickness of the <strong>top and bottom<\/strong> is <strong>twice<\/strong> that of the <strong>sides<\/strong>, so minimizing cost means optimizing the use of material rather than just minimizing total surface area.<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Volume Constraint<\/strong><br>The volume of a cylindrical can is given by:<br>[<br>V = \\pi r^2 h<br>]<br>Given that the can must hold <strong>12 fluid ounces<\/strong> (\u2248 21.66 in\u00b3), we set up the constraint:<br>[<br>\\pi r^2 h = 21.66<br>]<\/li>\n\n\n\n<li><strong>Material Cost Function<\/strong><\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The <strong>side surface area<\/strong> is ( 2\\pi rh ), with thickness ( t ).<\/li>\n\n\n\n<li>The <strong>top and bottom areas<\/strong> are each ( \\pi r^2 ), but their thickness is ( 2t ).<\/li>\n\n\n\n<li>Total cost is proportional to:<br>[<br>C = 2t(2\\pi rh) + 2(2t)(\\pi r^2)<br>]<br>[<br>C = 4\\pi r h t + 4\\pi r^2 t<br>]<br>[<br>C = 4\\pi r t (h + r)<br>]<br>Using the volume constraint to express ( h ) in terms of ( r ):<br>[<br>h = \\frac{21.66}{\\pi r^2}<br>]<br>Substituting into ( C ):<br>[<br>C = 4\\pi r t \\left( \\frac{21.66}{\\pi r^2} + r \\right)<br>]<br>[<br>C = 4t(21.66\/r + \\pi r^2)<br>]<br>To minimize cost, differentiate and solve ( dC\/dr = 0 ), yielding:<br>[<br>r = \\sqrt[3]{\\frac{21.66}{2\\pi}} \\approx 1.54 \\text{ in}, \\quad h = \\frac{21.66}{\\pi(1.54)^2} \\approx 2.91 \\text{ in}<br>]<\/li>\n<\/ul>\n\n\n\n<p>Thus, a <strong>radius of 1.54 inches and a height of 2.91 inches<\/strong> minimizes material cost.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A soda can is to hold 12 fluid ounces. Suppose that the bottom and top are twice as thick as the sides. Find the dimensions of the can that minimize the amount of material used. (Hint: Instead of minimizing surface area, minimize the cost, which is proportional to the product of the thickness and the [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-187005","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/187005","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=187005"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/187005\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=187005"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=187005"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=187005"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}