{"id":3059,"date":"2021-05-29T10:00:32","date_gmt":"2021-05-29T15:00:32","guid":{"rendered":"https:\/\/laurentlessard.com\/bookproofs\/?p=3059"},"modified":"2021-05-29T10:08:36","modified_gmt":"2021-05-29T15:08:36","slug":"tetrahedron-optimization","status":"publish","type":"post","link":"https:\/\/laurentlessard.com\/bookproofs\/tetrahedron-optimization\/","title":{"rendered":"Tetrahedron optimization"},"content":{"rendered":"

<\/p>This week\u2019s Riddler Classic<\/a> is a short problem 3D geometry. Here we go! (I paraphrased the question)\n

\nA polyhedron has six edges. Five of the edges have length $1$. What is the largest possible volume?\n<\/p><\/blockquote>\n

Here is my solution
\n[Show Solution]<\/a><\/p>\n

\n

The only polyhedron with six edges is a tetrahedron<\/a>, which is a pyramid with a triangular base. Two of the faces will be equilateral triangles that share a common edge. This accounts for the five edges of length 1. The length of the sixth edge is determined by the angle between the faces, which we will call $\\theta$. Here is an animation showing the different tetrahedra you get as you vary $\\theta$:<\/p>\n

\"\"<\/a><\/p>\n

In this diagram, $AB=BC=AC=AD=BD=1$ and $OD=OC=\\frac{\\sqrt{3}}{2}$.<\/p>\n

The volume is equal to
\n\\begin{align}
\nV&=\\frac{1}{3}(\\text{Area of base})\\cdot(\\text{altitude}) \\\\
\n&= \\frac{1}{3}(\\text{Area ABC})\\cdot(DG) \\\\
\n&= \\frac{1}{3}\\left( \\frac{1}{2} (AB)(OC) \\right) \\cdot \\left( (OD) \\sin\\theta \\right) \\\\
\n&= \\frac{1}{3} \\cdot \\frac{1}{2}\\cdot 1 \\cdot \\frac{\\sqrt{3}}{2} \\cdot \\frac{\\sqrt{3}}{2} \\cdot \\sin\\theta \\\\
\n&= \\frac{1}{8}\\cdot\\sin\\theta
\n\\end{align}Therefore, the maximum volume is $\\frac{1}{8}$ and it occurs when $\\theta=90^\\circ$. This is intuitive because the area of the base is fixed, so the largest volume occurs when the altitude is as large as possible.\n<\/p><\/div>\n

<\/p>\n<\/body>","protected":false},"excerpt":{"rendered":"

This week\u2019s Riddler Classic is a short problem 3D geometry. Here we go! (I paraphrased the question) A polyhedron has six edges. Five of the edges have length $1$. What is the largest possible volume? Here is my solution [Show Solution] The only polyhedron with six edges is a tetrahedron, which is a pyramid with … Continue reading “Tetrahedron optimization”<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":3060,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"_uf_show_specific_survey":0,"_uf_disable_surveys":false,"footnotes":""},"categories":[7],"tags":[10,26],"class_list":["post-3059","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-riddler","tag-geometry","tag-optimization"],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t