{"id":3252,"date":"2025-08-15T23:46:11","date_gmt":"2025-08-15T23:46:11","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=3252"},"modified":"2026-03-24T17:40:17","modified_gmt":"2026-03-24T17:40:17","slug":"modeling-with-trigonometric-equations-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/modeling-with-trigonometric-equations-apply-it\/","title":{"raw":"Modeling with Trigonometric Equations: Apply It","rendered":"Modeling with Trigonometric Equations: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Determine the amplitude and period of a periodic context<\/li>\r\n \t<li>Model periodic behavior with sinusoidal functions<\/li>\r\n \t<li>Write both a sine and cosine function to model the same periodic behavior<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Temperature Variation in a Desert City<\/h2>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-normal break-words\">The temperature in Phoenix, Arizona varies throughout the day. On a typical summer day, the temperature reaches a maximum of 105\u00b0F at 4:00 PM and a minimum of 75\u00b0F at 4:00 AM. The pattern repeats every 24 hours.<\/p>\r\n\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li class=\"whitespace-normal break-words\">Find a sinusoidal function of the form [latex]T(t) = A\\cos(B(t - C)) + D[\/latex] that models the temperature, where [latex]t[\/latex] is hours after midnight.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">What is the temperature at 10:00 AM?<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Write an equivalent sine function to model the same temperature pattern.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"973777\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"973777\"]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>The cosine function:\r\n<p class=\"whitespace-pre-wrap break-words\">Calculate amplitude: [latex]A = \\frac{105 - 75}{2} = 15[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Calculate vertical shift: [latex]D = \\frac{105 + 75}{2} = 90[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Calculate [latex]B[\/latex] from the 24-hour period: [latex]B = \\frac{2\\pi}{24} = \\frac{\\pi}{12}[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">Find phase shift: Since cosine starts at its maximum and the maximum temperature occurs at [latex]t = 16[\/latex] (4:00 PM), we have [latex]C = 16[\/latex].<\/p>\r\n<\/li>\r\n \t<li>The temperature at 10:00am\r\n<p class=\"whitespace-normal break-words\">At 10:00 AM, [latex]t = 10[\/latex]:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[latex]\\begin{aligned} T(10) &amp;= 15\\cos\\left(\\frac{\\pi}{12}(10 - 16)\\right) + 90 \\ &amp;= 15\\cos\\left(\\frac{\\pi}{12}(-6)\\right) + 90 \\ &amp;= 15\\cos\\left(-\\frac{\\pi}{2}\\right) + 90 \\ &amp;= 15(0) + 90 \\ &amp;= 90\u00b0\\text{F} \\end{aligned}[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">The temperature at 10:00 AM is 90\u00b0F (the average temperature).<\/p>\r\n<\/li>\r\n \t<li>The sine function\r\n<p class=\"whitespace-normal break-words\">Sine reaches its maximum [latex]\\frac{1}{4}[\/latex] period after it crosses the midline going upward. Since our period is 24 hours, [latex]\\frac{1}{4}[\/latex] period = 6 hours.<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">The maximum occurs at [latex]t = 16[\/latex], so sine crosses the midline going up at: [latex]16 - 6 = 10[\/latex]<\/p>\r\n[latex]T(t) = 15\\sin\\left(\\frac{\\pi}{12}(t - 10)\\right) + 90[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">Choose cosine when your reference point is at a maximum or minimum. Choose sine when your reference point is at the midline. This makes finding the phase shift much easier!<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p class=\"whitespace-normal break-words\">The water depth at a dock varies with the tides. High tide of 12 feet occurs at noon, and low tide of 4 feet occurs at 6:00 PM. The cycle repeats every 12 hours.<\/p>\r\n\r\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Find a cosine function [latex]d(t) = A\\cos(B(t - C)) + D[\/latex] to model the depth, where [latex]t[\/latex] is hours after midnight<\/li>\r\n \t<li class=\"whitespace-normal break-words\">What is the water depth at 3:00 PM?<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Write an equivalent sine function<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p class=\"whitespace-normal break-words\">A weight on a spring oscillates vertically with position given by [latex]y = 7\\cos\\left(\\frac{\\pi}{3}t\\right)[\/latex] cm, where [latex]t[\/latex] is time in seconds. Find:<\/p>\r\n\r\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">The maximum displacement of the weight<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The period of oscillation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The frequency<\/li>\r\n \t<li class=\"whitespace-normal break-words\">At what time during the first period does the weight first reach [latex]y = 3.5[\/latex] cm.<\/li>\r\n<\/ol>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Determine the amplitude and period of a periodic context<\/li>\n<li>Model periodic behavior with sinusoidal functions<\/li>\n<li>Write both a sine and cosine function to model the same periodic behavior<\/li>\n<\/ul>\n<\/section>\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Temperature Variation in a Desert City<\/h2>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-normal break-words\">The temperature in Phoenix, Arizona varies throughout the day. On a typical summer day, the temperature reaches a maximum of 105\u00b0F at 4:00 PM and a minimum of 75\u00b0F at 4:00 AM. The pattern repeats every 24 hours.<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li class=\"whitespace-normal break-words\">Find a sinusoidal function of the form [latex]T(t) = A\\cos(B(t - C)) + D[\/latex] that models the temperature, where [latex]t[\/latex] is hours after midnight.<\/li>\n<li class=\"whitespace-normal break-words\">What is the temperature at 10:00 AM?<\/li>\n<li class=\"whitespace-normal break-words\">Write an equivalent sine function to model the same temperature pattern.<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q973777\">Show Solution<\/button><\/p>\n<div id=\"q973777\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li>The cosine function:\n<p class=\"whitespace-pre-wrap break-words\">Calculate amplitude: [latex]A = \\frac{105 - 75}{2} = 15[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Calculate vertical shift: [latex]D = \\frac{105 + 75}{2} = 90[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Calculate [latex]B[\/latex] from the 24-hour period: [latex]B = \\frac{2\\pi}{24} = \\frac{\\pi}{12}[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">Find phase shift: Since cosine starts at its maximum and the maximum temperature occurs at [latex]t = 16[\/latex] (4:00 PM), we have [latex]C = 16[\/latex].<\/p>\n<\/li>\n<li>The temperature at 10:00am\n<p class=\"whitespace-normal break-words\">At 10:00 AM, [latex]t = 10[\/latex]:<\/p>\n<p class=\"whitespace-pre-wrap break-words\">[latex]\\begin{aligned} T(10) &= 15\\cos\\left(\\frac{\\pi}{12}(10 - 16)\\right) + 90 \\ &= 15\\cos\\left(\\frac{\\pi}{12}(-6)\\right) + 90 \\ &= 15\\cos\\left(-\\frac{\\pi}{2}\\right) + 90 \\ &= 15(0) + 90 \\ &= 90\u00b0\\text{F} \\end{aligned}[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">The temperature at 10:00 AM is 90\u00b0F (the average temperature).<\/p>\n<\/li>\n<li>The sine function\n<p class=\"whitespace-normal break-words\">Sine reaches its maximum [latex]\\frac{1}{4}[\/latex] period after it crosses the midline going upward. Since our period is 24 hours, [latex]\\frac{1}{4}[\/latex] period = 6 hours.<\/p>\n<p class=\"whitespace-pre-wrap break-words\">The maximum occurs at [latex]t = 16[\/latex], so sine crosses the midline going up at: [latex]16 - 6 = 10[\/latex]<\/p>\n<p>[latex]T(t) = 15\\sin\\left(\\frac{\\pi}{12}(t - 10)\\right) + 90[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">Choose cosine when your reference point is at a maximum or minimum. Choose sine when your reference point is at the midline. This makes finding the phase shift much easier!<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p class=\"whitespace-normal break-words\">The water depth at a dock varies with the tides. High tide of 12 feet occurs at noon, and low tide of 4 feet occurs at 6:00 PM. The cycle repeats every 12 hours.<\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Find a cosine function [latex]d(t) = A\\cos(B(t - C)) + D[\/latex] to model the depth, where [latex]t[\/latex] is hours after midnight<\/li>\n<li class=\"whitespace-normal break-words\">What is the water depth at 3:00 PM?<\/li>\n<li class=\"whitespace-normal break-words\">Write an equivalent sine function<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p class=\"whitespace-normal break-words\">A weight on a spring oscillates vertically with position given by [latex]y = 7\\cos\\left(\\frac{\\pi}{3}t\\right)[\/latex] cm, where [latex]t[\/latex] is time in seconds. Find:<\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">The maximum displacement of the weight<\/li>\n<li class=\"whitespace-normal break-words\">The period of oscillation<\/li>\n<li class=\"whitespace-normal break-words\">The frequency<\/li>\n<li class=\"whitespace-normal break-words\">At what time during the first period does the weight first reach [latex]y = 3.5[\/latex] cm.<\/li>\n<\/ol>\n<\/section>\n","protected":false},"author":67,"menu_order":38,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":201,"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\/3252"}],"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\/67"}],"version-history":[{"count":4,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3252\/revisions"}],"predecessor-version":[{"id":5995,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3252\/revisions\/5995"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/201"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3252\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=3252"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=3252"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=3252"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=3252"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}