{"id":4211,"date":"2025-01-04T08:59:01","date_gmt":"2025-01-04T13:59:01","guid":{"rendered":"https:\/\/josmfs.net\/?p=4211"},"modified":"2025-01-04T09:03:57","modified_gmt":"2025-01-04T14:03:57","slug":"double-areas-puzzles","status":"publish","type":"post","link":"https:\/\/josmfs.net\/wordpress\/2025\/01\/04\/double-areas-puzzles\/","title":{"rendered":"Double Areas Puzzles"},"content":{"rendered":"<p><img loading=\"lazy\" decoding=\"async\" class=\"size-large wp-image-4209 aligncenter\" src=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2025\/01\/Double-Areas-Fig2-1024x273.jpg\" alt=\"\" width=\"625\" height=\"167\" srcset=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2025\/01\/Double-Areas-Fig2-1024x273.jpg 1024w, https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2025\/01\/Double-Areas-Fig2-300x80.jpg 300w, https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2025\/01\/Double-Areas-Fig2-768x205.jpg 768w, https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2025\/01\/Double-Areas-Fig2-624x166.jpg 624w, https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2025\/01\/Double-Areas-Fig2.jpg 1200w\" sizes=\"auto, (max-width: 625px) 100vw, 625px\" \/>A while ago James Tanton provided a series of puzzles:<\/p>\n<p><strong><a href=\"https:\/\/twitter.com\/jamestanton\/status\/1606239945924001793\">Puzzle #1<\/a><\/strong> \u00a0 At what value between 0 and 1 does a horizontal line at that height produce <strong>two regions of equal area<\/strong> as shown on the graph of <em>y = x<\/em><sup>2<\/sup>?<\/p>\n<p><strong><a href=\"https:\/\/twitter.com\/jamestanton\/status\/1607327864529059843\">Puzzle #2<\/a><\/strong> \u00a0 A horizontal line is drawn between the lines <em>y<\/em> = 0 and <em>y<\/em> = 1, dividing the graph of <em>y = x<\/em><sup>2<\/sup> into two regions as shown. At what height should that line be drawn so that the <strong>sum of the areas of these two regions is minimal<\/strong>?<\/p>\n<p><strong><a href=\"https:\/\/twitter.com\/jamestanton\/status\/1607679053762371585\">Puzzle #3<\/a><\/strong> \u00a0 A horizontal line is drawn between the lines y = 0 and y = 1, dividing the graph of <em>y = x<sup>n<\/sup><\/em> into two regions as shown (<em>n<\/em> &gt; 0). At what height should that line be drawn so that the <strong>sum of the areas<\/strong> <strong>of these two regions is minimal<\/strong>? Does that height depend on the value of <em>n<\/em>?<\/p>\n<p><strong><a href=\"https:\/\/twitter.com\/jamestanton\/status\/1608046851864788992\">Puzzle #4<\/a><\/strong> \u00a0 What horizontal line drawn between <em>y<\/em> = 0 and <em>y<\/em> = 1 on the graph of <em>y<\/em> = 2<sup>\u221a<em>x<\/em><\/sup> \u2013 1 <strong>minimizes the sum of the two shaded areas<\/strong> shown?<\/p>\n<p>See <a href=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2025\/01\/Double-Areas-Puzzles-241231.pdf\">Double Areas Puzzles<\/a> for solutions.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A while ago James Tanton provided a series of puzzles: Puzzle #1 \u00a0 At what value between 0 and 1 does a horizontal line at that height produce two regions of equal area as shown on the graph of y = x2? Puzzle #2 \u00a0 A horizontal line is drawn between the lines y = [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[32,43,153],"class_list":["post-4211","post","type-post","status-publish","format-standard","hentry","category-math-inquiries","tag-calculus","tag-james-tanton","tag-optimization"],"_links":{"self":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/4211","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/comments?post=4211"}],"version-history":[{"count":1,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/4211\/revisions"}],"predecessor-version":[{"id":4212,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/4211\/revisions\/4212"}],"wp:attachment":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/media?parent=4211"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/categories?post=4211"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/tags?post=4211"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}