{"id":321,"date":"2019-01-01T16:41:03","date_gmt":"2019-01-01T21:41:03","guid":{"rendered":"http:\/\/josmfs.net\/wordpress\/?p=321"},"modified":"2024-04-15T15:00:58","modified_gmt":"2024-04-15T19:00:58","slug":"triangular-boundary-problem","status":"publish","type":"post","link":"https:\/\/josmfs.net\/wordpress\/2019\/01\/01\/triangular-boundary-problem\/","title":{"rendered":"Triangular Boundary Problem"},"content":{"rendered":"<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft  wp-image-3936\" src=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2024\/04\/Triangular-Boundary-Prob-Fig2.jpg\" alt=\"\" width=\"250\" height=\"85\" srcset=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2024\/04\/Triangular-Boundary-Prob-Fig2.jpg 500w, https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2024\/04\/Triangular-Boundary-Prob-Fig2-300x102.jpg 300w\" sizes=\"auto, (max-width: 250px) 100vw, 250px\" \/>This problem comes from the defunct <em>Wall Street Journal<\/em> Varsity Math Week collection.<\/p>\n<p>\u201cThe coach then shows the team the diagram to the left and asks: What is the maximum area of a rectangle contained entirely within a triangle with sides of 9, 10 and 17?\u201d<\/p>\n<p>I changed the numbers a bit to make my calculations easier, but left the problem otherwise unchanged. When I checked the Varsity Math Week solution, I saw they used a simplifying formula that I could not remember. I also believed their solution left out a justification for the maximal area. Besides an intuitive solution for this, I also included a calculus version.<\/p>\n<p><a href=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2024\/04\/Triangular-Boundary-Problem-Answer.jpg\">Answer<\/a>.<\/p>\n<p>See the <a href=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2019\/01\/Triangular-Boundary-Problem-171003.pdf\">Triangular Boundary Problem<\/a> for solutions.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>This problem comes from the defunct Wall Street Journal Varsity Math Week collection. \u201cThe coach then shows the team the diagram to the left and asks: What is the maximum area of a rectangle contained entirely within a triangle with sides of 9, 10 and 17?\u201d I changed the numbers a bit to make my [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[32,13,68],"class_list":["post-321","post","type-post","status-publish","format-standard","hentry","category-puzzles-and-problems","tag-calculus","tag-plane-geometry","tag-varsity-math-week"],"_links":{"self":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/321","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=321"}],"version-history":[{"count":4,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/321\/revisions"}],"predecessor-version":[{"id":3938,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/321\/revisions\/3938"}],"wp:attachment":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/media?parent=321"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/categories?post=321"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/tags?post=321"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}