{"id":2444,"date":"2016-11-03T22:20:22","date_gmt":"2016-11-03T22:20:22","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebra\/?post_type=chapter&#038;p=2444"},"modified":"2017-04-04T19:58:42","modified_gmt":"2017-04-04T19:58:42","slug":"putting-it-together-conic-sections","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/chapter\/putting-it-together-conic-sections\/","title":{"raw":"Putting It Together: Conic Sections","rendered":"Putting It Together: Conic Sections"},"content":{"raw":"At the beginning of this module, you were presented with the challenge of creating a whispering gallery. \u00a0Now that you know about conics, you can do just this. \u00a0It turns out that a whispering gallery forms under an elliptical arch.\r\n\r\n<img class=\" wp-image-3653 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/03\/16203832\/vaulted-cellar-247391_1920-300x199.jpg\" alt=\"Series of brick archways at the base of a building.\" width=\"455\" height=\"302\" \/>\r\n\r\n&nbsp;\r\n\r\nYou can create the gallery by placing the whispering dishes at the foci of the ellipse. \u00a0Let\u2019s place the center of the arch above the origin. \u00a0The equation for an ellipse centered at the origin is\r\n<p style=\"text-align: center;\">[latex]{\\Large\\frac{x^2}{a^2}+\\frac{y^2}{b^2}}=1[\/latex]<\/p>\r\nThe width of the arch is 100 feet, so [latex]a=50[\/latex]. The height of the arch is 40 feet, so [latex]b=40[\/latex]. \u00a0Now you can rewrite the ellipse equation.\r\n<p style=\"text-align: center;\">[latex]{\\Large\\frac{x^2}{2500}}+{\\Large\\frac{y^2}{1600}}=1[\/latex]<\/p>\r\nTo find the distance, [latex]c[\/latex],\u00a0from the origin to the foci of the ellipse, use\r\n<p style=\"text-align: center;\">[latex]c^2=a^2-b^2[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]c^2=50^2-40^2[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]c^2=900[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]c=30[\/latex]<\/p>\r\nSo you need to place the whispering dishes 30 feet from the center of the room.\r\n\r\n<iframe style=\"border: 1px solid #ccc;\" src=\"https:\/\/www.desmos.com\/calculator\/yq28h7pc2g?embed\" width=\"500px\" height=\"500px\" frameborder=\"0\"><\/iframe>\r\n\r\n&nbsp;\r\n\r\nWhat if the height of the arch were lowered to 30 feet? \u00a0Could you still make it a whispering gallery?\r\n\r\nIf this happens, the width of the arch is still 100 feet, so [latex]a=50[\/latex]. \u00a0But the height of the arch is 30 feet, so [latex]b=30[\/latex]. \u00a0Now you can rewrite the ellipse equation.\r\n<p style=\"text-align: center;\">[latex]{\\Large\\frac{x^2}{2500}}+{\\Large\\frac{y^2}{900}}=1[\/latex]<\/p>\r\n&nbsp;\r\n\r\nTo find the distance, c, from the origin to the foci of the ellipse, use\r\n<p style=\"text-align: center;\">[latex]c^2=a^2-b^2[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]c^2=50^2-30^2[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]c^2=1600[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]c=40[\/latex]<\/p>\r\nYou\u2019ll have to move the dishes out another 10 feet on each side so that they are 40 feet from the center. \u00a0Thanks to your knowledge of conics, you can use the dimensions of the arch to place the dishes where they belong to create a whispering gallery. \u00a0Now, just be careful not to share any secrets you don\u2019t want anyone to hear.\r\n\r\n&nbsp;\r\n\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/oerfiles\/College+Algebra\/calculator.html\" target=\"_blank\"><img class=\"alignnone size-full wp-image-3370\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/13193222\/calculator.png\" alt=\"\" width=\"251\" height=\"46\" \/><\/a>","rendered":"<p>At the beginning of this module, you were presented with the challenge of creating a whispering gallery. \u00a0Now that you know about conics, you can do just this. \u00a0It turns out that a whispering gallery forms under an elliptical arch.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3653 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/03\/16203832\/vaulted-cellar-247391_1920-300x199.jpg\" alt=\"Series of brick archways at the base of a building.\" width=\"455\" height=\"302\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>You can create the gallery by placing the whispering dishes at the foci of the ellipse. \u00a0Let\u2019s place the center of the arch above the origin. \u00a0The equation for an ellipse centered at the origin is<\/p>\n<p style=\"text-align: center;\">[latex]{\\Large\\frac{x^2}{a^2}+\\frac{y^2}{b^2}}=1[\/latex]<\/p>\n<p>The width of the arch is 100 feet, so [latex]a=50[\/latex]. The height of the arch is 40 feet, so [latex]b=40[\/latex]. \u00a0Now you can rewrite the ellipse equation.<\/p>\n<p style=\"text-align: center;\">[latex]{\\Large\\frac{x^2}{2500}}+{\\Large\\frac{y^2}{1600}}=1[\/latex]<\/p>\n<p>To find the distance, [latex]c[\/latex],\u00a0from the origin to the foci of the ellipse, use<\/p>\n<p style=\"text-align: center;\">[latex]c^2=a^2-b^2[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]c^2=50^2-40^2[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]c^2=900[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]c=30[\/latex]<\/p>\n<p>So you need to place the whispering dishes 30 feet from the center of the room.<\/p>\n<p><iframe loading=\"lazy\" style=\"border: 1px solid #ccc;\" src=\"https:\/\/www.desmos.com\/calculator\/yq28h7pc2g?embed\" width=\"500px\" height=\"500px\" frameborder=\"0\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<p>What if the height of the arch were lowered to 30 feet? \u00a0Could you still make it a whispering gallery?<\/p>\n<p>If this happens, the width of the arch is still 100 feet, so [latex]a=50[\/latex]. \u00a0But the height of the arch is 30 feet, so [latex]b=30[\/latex]. \u00a0Now you can rewrite the ellipse equation.<\/p>\n<p style=\"text-align: center;\">[latex]{\\Large\\frac{x^2}{2500}}+{\\Large\\frac{y^2}{900}}=1[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>To find the distance, c, from the origin to the foci of the ellipse, use<\/p>\n<p style=\"text-align: center;\">[latex]c^2=a^2-b^2[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]c^2=50^2-30^2[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]c^2=1600[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]c=40[\/latex]<\/p>\n<p>You\u2019ll have to move the dishes out another 10 feet on each side so that they are 40 feet from the center. \u00a0Thanks to your knowledge of conics, you can use the dimensions of the arch to place the dishes where they belong to create a whispering gallery. \u00a0Now, just be careful not to share any secrets you don\u2019t want anyone to hear.<\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/oerfiles\/College+Algebra\/calculator.html\" target=\"_blank\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-3370\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/13193222\/calculator.png\" alt=\"\" width=\"251\" height=\"46\" \/><\/a><\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-2444\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Tunnel Arches. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/pixabay.com\/en\/vaulted-cellar-tunnel-arches-keller-247391\">https:\/\/pixabay.com\/en\/vaulted-cellar-tunnel-arches-keller-247391<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/cc0\">CC0: No Rights Reserved<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":21,"menu_order":14,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\"},{\"type\":\"cc\",\"description\":\"Tunnel Arches\",\"author\":\"\",\"organization\":\"\",\"url\":\"https:\/\/pixabay.com\/en\/vaulted-cellar-tunnel-arches-keller-247391\",\"project\":\"\",\"license\":\"cc0\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"dfb21ae7-d2a4-4024-84f3-869b069d587f","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2444","chapter","type-chapter","status-publish","hentry"],"part":2320,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2444","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/wp\/v2\/users\/21"}],"version-history":[{"count":11,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2444\/revisions"}],"predecessor-version":[{"id":3936,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2444\/revisions\/3936"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/2320"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2444\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/wp\/v2\/media?parent=2444"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2444"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/wp\/v2\/contributor?post=2444"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/wp\/v2\/license?post=2444"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}