{"id":2551,"date":"2025-08-13T18:02:41","date_gmt":"2025-08-13T18:02:41","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=2551"},"modified":"2025-10-22T23:16:23","modified_gmt":"2025-10-22T23:16:23","slug":"parabolas-apply-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/parabolas-apply-it-1\/","title":{"raw":"Parabolas: Apply It 1","rendered":"Parabolas: Apply It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Identify the vertex, focus, directrix, and endpoints of the latus rectum.<\/li>\r\n \t<li>Write equations of parabolas in standard form.<\/li>\r\n \t<li>Graph parabolas.<\/li>\r\n \t<li>Solve applied problems involving parabolas.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Solving Applied Problems Involving Parabolas<\/h2>\r\nAs we mentioned at the beginning of the section, parabolas are used to design many objects we use every day, such as telescopes, suspension bridges, microphones, and radar equipment. Parabolic mirrors, such as the one used to light the Olympic torch, have a very unique reflecting property. When rays of light parallel to the parabola\u2019s <strong>axis of symmetry<\/strong> are directed toward any surface of the mirror, the light is reflected directly to the focus.\u00a0This is why the Olympic torch is ignited when it is held at the focus of the parabolic mirror.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27182941\/CNX_Precalc_Figure_10_03_0142.jpg\" alt=\"\" width=\"487\" height=\"362\" \/> Reflecting property of parabolas[\/caption]\r\n\r\nParabolic mirrors have the ability to focus the sun\u2019s energy to a single point, raising the temperature hundreds of degrees in a matter of seconds. Thus, parabolic mirrors are featured in many low-cost, energy efficient solar products, such as solar cookers, solar heaters, and even travel-sized fire starters.\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">A cross-section of a design for a travel-sized solar fire starter is shown in Figure 13. The sun\u2019s rays reflect off the parabolic mirror toward an object attached to the igniter. Because the igniter is located at the focus of the parabola, the reflected rays cause the object to burn in just seconds.\r\n<ol>\r\n \t<li>Find the equation of the parabola that models the fire starter. Assume that the vertex of the parabolic mirror is the origin of the coordinate plane.<\/li>\r\n \t<li>Use the equation found in part (a) to find the depth of the fire starter.<\/li>\r\n<\/ol>\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27182943\/CNX_Precalc_Figure_10_03_0162.jpg\" alt=\"\" width=\"487\" height=\"217\" \/> Cross-section of a travel-sized solar fire starter[\/caption]\r\n\r\n[reveal-answer q=\"430262\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"430262\"]\r\n<ol>\r\n \t<li>The vertex of the dish is the origin of the coordinate plane, so the parabola will take the standard form [latex]{x}^{2}=4py[\/latex], where [latex]p&gt;0[\/latex]. The igniter, which is the focus, is 1.7 inches above the vertex of the dish. Thus we have [latex]p=1.7[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{align}&amp;{x}^{2}=4py &amp;&amp;\\text{Standard form of upward-facing parabola with vertex (0,0)} \\\\ &amp;{x}^{2}=4\\left(1.7\\right)y &amp;&amp;\\text{Substitute 1}\\text{.7 for }p. \\\\ &amp;{x}^{2}=6.8y &amp;&amp;\\text{Multiply}. \\end{align}[\/latex]<\/div><\/li>\r\n \t<li>The dish extends [latex]\\frac{4.5}{2}=2.25[\/latex] inches on either side of the origin. We can substitute 2.25 for [latex]x[\/latex] in the equation from part (a) to find the depth of the dish.\r\n<div style=\"text-align: center;\">[latex]\\begin{align}&amp;{x}^{2}=6.8y&amp;&amp; \\text{Equation found in part (a)}. \\\\ &amp;{\\left(2.25\\right)}^{2}=6.8y&amp;&amp; \\text{Substitute 2}\\text{.25 for }x. \\\\ &amp;y\\approx 0.74 &amp;&amp; \\text{Solve for }y. \\end{align}[\/latex]<\/div>\r\nThe dish is about 0.74 inches deep.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">Balcony-sized solar cookers have been designed for families living in India. The top of a dish has a diameter of 1600 mm. The sun\u2019s rays reflect off the parabolic mirror toward the \"cooker,\" which is placed 320 mm from the base.\r\n<p style=\"padding-left: 60px;\">a. Find an equation that models a cross-section of the solar cooker. Assume that the vertex of the parabolic mirror is the origin of the coordinate plane, and that the parabola opens to the right (i.e., has the <em>x<\/em>-axis as its axis of symmetry).<\/p>\r\n<p style=\"padding-left: 60px;\">b. Use the equation found in part (a) to find the depth of the cooker.<\/p>\r\n[reveal-answer q=\"92855\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"92855\"]\r\n\r\na.\u00a0[latex]{y}^{2}=1280x[\/latex]\r\nb. The depth of the cooker is 500 mm\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]87083[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Identify the vertex, focus, directrix, and endpoints of the latus rectum.<\/li>\n<li>Write equations of parabolas in standard form.<\/li>\n<li>Graph parabolas.<\/li>\n<li>Solve applied problems involving parabolas.<\/li>\n<\/ul>\n<\/section>\n<h2>Solving Applied Problems Involving Parabolas<\/h2>\n<p>As we mentioned at the beginning of the section, parabolas are used to design many objects we use every day, such as telescopes, suspension bridges, microphones, and radar equipment. Parabolic mirrors, such as the one used to light the Olympic torch, have a very unique reflecting property. When rays of light parallel to the parabola\u2019s <strong>axis of symmetry<\/strong> are directed toward any surface of the mirror, the light is reflected directly to the focus.\u00a0This is why the Olympic torch is ignited when it is held at the focus of the parabolic mirror.<\/p>\n<figure style=\"width: 487px\" 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\/27182941\/CNX_Precalc_Figure_10_03_0142.jpg\" alt=\"\" width=\"487\" height=\"362\" \/><figcaption class=\"wp-caption-text\">Reflecting property of parabolas<\/figcaption><\/figure>\n<p>Parabolic mirrors have the ability to focus the sun\u2019s energy to a single point, raising the temperature hundreds of degrees in a matter of seconds. Thus, parabolic mirrors are featured in many low-cost, energy efficient solar products, such as solar cookers, solar heaters, and even travel-sized fire starters.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">A cross-section of a design for a travel-sized solar fire starter is shown in Figure 13. The sun\u2019s rays reflect off the parabolic mirror toward an object attached to the igniter. Because the igniter is located at the focus of the parabola, the reflected rays cause the object to burn in just seconds.<\/p>\n<ol>\n<li>Find the equation of the parabola that models the fire starter. Assume that the vertex of the parabolic mirror is the origin of the coordinate plane.<\/li>\n<li>Use the equation found in part (a) to find the depth of the fire starter.<\/li>\n<\/ol>\n<figure style=\"width: 487px\" 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\/27182943\/CNX_Precalc_Figure_10_03_0162.jpg\" alt=\"\" width=\"487\" height=\"217\" \/><figcaption class=\"wp-caption-text\">Cross-section of a travel-sized solar fire starter<\/figcaption><\/figure>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q430262\">Show Solution<\/button><\/p>\n<div id=\"q430262\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>The vertex of the dish is the origin of the coordinate plane, so the parabola will take the standard form [latex]{x}^{2}=4py[\/latex], where [latex]p>0[\/latex]. The igniter, which is the focus, is 1.7 inches above the vertex of the dish. Thus we have [latex]p=1.7[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{align}&{x}^{2}=4py &&\\text{Standard form of upward-facing parabola with vertex (0,0)} \\\\ &{x}^{2}=4\\left(1.7\\right)y &&\\text{Substitute 1}\\text{.7 for }p. \\\\ &{x}^{2}=6.8y &&\\text{Multiply}. \\end{align}[\/latex]<\/div>\n<\/li>\n<li>The dish extends [latex]\\frac{4.5}{2}=2.25[\/latex] inches on either side of the origin. We can substitute 2.25 for [latex]x[\/latex] in the equation from part (a) to find the depth of the dish.\n<div style=\"text-align: center;\">[latex]\\begin{align}&{x}^{2}=6.8y&& \\text{Equation found in part (a)}. \\\\ &{\\left(2.25\\right)}^{2}=6.8y&& \\text{Substitute 2}\\text{.25 for }x. \\\\ &y\\approx 0.74 && \\text{Solve for }y. \\end{align}[\/latex]<\/div>\n<p>The dish is about 0.74 inches deep.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">Balcony-sized solar cookers have been designed for families living in India. The top of a dish has a diameter of 1600 mm. The sun\u2019s rays reflect off the parabolic mirror toward the &#8220;cooker,&#8221; which is placed 320 mm from the base.<\/p>\n<p style=\"padding-left: 60px;\">a. Find an equation that models a cross-section of the solar cooker. Assume that the vertex of the parabolic mirror is the origin of the coordinate plane, and that the parabola opens to the right (i.e., has the <em>x<\/em>-axis as its axis of symmetry).<\/p>\n<p style=\"padding-left: 60px;\">b. Use the equation found in part (a) to find the depth of the cooker.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q92855\">Show Solution<\/button><\/p>\n<div id=\"q92855\" class=\"hidden-answer\" style=\"display: none\">\n<p>a.\u00a0[latex]{y}^{2}=1280x[\/latex]<br \/>\nb. The depth of the cooker is 500 mm<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm87083\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=87083&theme=lumen&iframe_resize_id=ohm87083&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":13,"menu_order":20,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":522,"module-header":"apply_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2551"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/13"}],"version-history":[{"count":3,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2551\/revisions"}],"predecessor-version":[{"id":4842,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2551\/revisions\/4842"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/522"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2551\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2551"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2551"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2551"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2551"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}