{"id":17954,"date":"2021-09-11T19:32:25","date_gmt":"2021-09-11T19:32:25","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/?post_type=chapter&#038;p=17954"},"modified":"2021-09-11T19:33:24","modified_gmt":"2021-09-11T19:33:24","slug":"10-1-conics","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/chapter\/10-1-conics\/","title":{"raw":"Section 10.1: Conics","rendered":"Section 10.1: Conics"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Know the names of the conics<\/li>\r\n<\/ul>\r\n<\/div>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"488\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27182703\/CNX_Precalc_Figure_10_01_001n2.jpg\" alt=\"A large ornate room with statues.\" width=\"488\" height=\"324\" \/> <b>Figure 1.<\/b> The National Statuary Hall in Washington, D.C. (credit: Greg Palmer, Flickr)[\/caption]\r\n\r\nCan you imagine standing at one end of a large room and still being able to hear a whisper from a person standing at the other end? The National Statuary Hall in Washington, D.C.\u00a0is such a room.[footnote]Architect of the Capitol. <a href=\"http:\/\/www.aoc.gov\" target=\"_blank\" rel=\"noopener\">http:\/\/www.aoc.gov<\/a>. Accessed April 15, 2014.[\/footnote]\u00a0It is an oval-shaped room called a <em>whispering chamber<\/em> because the shape makes it possible for sound to travel along the walls. In this section, we will investigate the shape of this room and its real-world applications, including how far apart two people in Statuary Hall can stand and still hear each other whisper.\r\n<h2>Know the Names of the Conics<\/h2>\r\nA conic section, or <strong>conic<\/strong>, is a shape resulting from intersecting a right circular cone with a plane. The angle at which the plane intersects the cone determines the shape.\u00a0 These conics are <strong>circles<\/strong> when the plane is perpendicular to the axis of the cone and intersects each generator, but intersects only one nappe of the cone; <strong>parabolas<\/strong> when the plane is tilted farther so that it is parallel to one (and only one) generator and intersects only one nappe of the cone; and <strong>hyperbolas<\/strong> when the plane intersects both nappes. See figure 2.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"976\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27182705\/CNX_Precalc_Figure_10_01_0022.jpg\" alt=\"\" width=\"976\" height=\"441\" \/> <b>Figure 2<\/b>[\/caption]\r\n\r\nConic sections can also be described by a set of points in the coordinate plane. Later in this chapter, we will see that the graph of any quadratic equation in two variables is a conic section. The signs of the equations and the coefficients of the variable terms determine the shape.","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Know the names of the conics<\/li>\n<\/ul>\n<\/div>\n<div style=\"width: 498px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27182703\/CNX_Precalc_Figure_10_01_001n2.jpg\" alt=\"A large ornate room with statues.\" width=\"488\" height=\"324\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 1.<\/b> The National Statuary Hall in Washington, D.C. (credit: Greg Palmer, Flickr)<\/p>\n<\/div>\n<p>Can you imagine standing at one end of a large room and still being able to hear a whisper from a person standing at the other end? The National Statuary Hall in Washington, D.C.\u00a0is such a room.<a class=\"footnote\" title=\"Architect of the Capitol. http:\/\/www.aoc.gov. Accessed April 15, 2014.\" id=\"return-footnote-17954-1\" href=\"#footnote-17954-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a>\u00a0It is an oval-shaped room called a <em>whispering chamber<\/em> because the shape makes it possible for sound to travel along the walls. In this section, we will investigate the shape of this room and its real-world applications, including how far apart two people in Statuary Hall can stand and still hear each other whisper.<\/p>\n<h2>Know the Names of the Conics<\/h2>\n<p>A conic section, or <strong>conic<\/strong>, is a shape resulting from intersecting a right circular cone with a plane. The angle at which the plane intersects the cone determines the shape.\u00a0 These conics are <strong>circles<\/strong> when the plane is perpendicular to the axis of the cone and intersects each generator, but intersects only one nappe of the cone; <strong>parabolas<\/strong> when the plane is tilted farther so that it is parallel to one (and only one) generator and intersects only one nappe of the cone; and <strong>hyperbolas<\/strong> when the plane intersects both nappes. See figure 2.<\/p>\n<div style=\"width: 986px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27182705\/CNX_Precalc_Figure_10_01_0022.jpg\" alt=\"\" width=\"976\" height=\"441\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 2<\/b><\/p>\n<\/div>\n<p>Conic sections can also be described by a set of points in the coordinate plane. Later in this chapter, we will see that the graph of any quadratic equation in two variables is a conic section. The signs of the equations and the coefficients of the variable terms determine the shape.<\/p>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-17954-1\">Architect of the Capitol. <a href=\"http:\/\/www.aoc.gov\" target=\"_blank\" rel=\"noopener\">http:\/\/www.aoc.gov<\/a>. Accessed April 15, 2014. <a href=\"#return-footnote-17954-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":264444,"menu_order":1,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-17954","chapter","type-chapter","status-publish","hentry"],"part":14618,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/17954","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/users\/264444"}],"version-history":[{"count":2,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/17954\/revisions"}],"predecessor-version":[{"id":17956,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/17954\/revisions\/17956"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/parts\/14618"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/17954\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/media?parent=17954"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapter-type?post=17954"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/contributor?post=17954"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/license?post=17954"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}