{"id":1355,"date":"2024-05-10T16:57:18","date_gmt":"2024-05-10T16:57:18","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=1355"},"modified":"2025-08-13T15:47:23","modified_gmt":"2025-08-13T15:47:23","slug":"applications-and-inequalities-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/applications-and-inequalities-learn-it-1\/","title":{"raw":"Applications of Non-Linear Equations: Learn It 1","rendered":"Applications of Non-Linear Equations: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li>Construct and apply non-linear equations and formulas to solve real-world problems.<\/li>\r\n<\/ul>\r\n<\/section>In many real-world situations, relationships between variables are not always straightforward or linear. Non-linear equations and formulas play a crucial role in accurately modeling and solving complex problems across various fields, including physics, engineering, economics, and environmental science.\r\n<p class=\"whitespace-pre-wrap break-words\">Non-linear equations are essential tools in our mathematical toolkit for several reasons:<\/p>\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Accurate Representation<\/strong>: Non-linear equations can capture the complexity of real-world phenomena more accurately than linear approximations. Many natural and man-made systems exhibit non-linear behavior, and using these equations allows us to model them faithfully.<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Diverse Applications<\/strong>: The applications of non-linear equations span numerous disciplines:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Physics: Describing planetary motion<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Biology: Modeling population growth<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Economics: Analyzing market trends<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Environmental Science: Predicting climate patterns<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Problem-Solving Power<\/strong>: By constructing and applying non-linear equations, we can:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Solve intricate problems<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Make predictions about complex system behavior<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Gain insights that might be impossible with simpler linear models<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<h2>Quadratic Applications<\/h2>\r\nProjectile motion happens when you throw a ball into the air and it comes back down because of gravity. \u00a0A projectile will follow a curved path that behaves in a predictable way. \u00a0This predictable motion has been studied for centuries, and in simple cases it\u2019s height from the ground\u00a0at a given time, [latex]t[\/latex], can be modeled with a quadratic polynomial of the form [latex]\\text{height} = at^2+bt+c[\/latex]. Projectile motion is also called a parabolic trajectory because of the shape of the path of a projectile\u2019s motion, as in the image of water in the fountain below.\r\n\r\n[caption id=\"attachment_1356\" align=\"aligncenter\" width=\"225\"]<img class=\"wp-image-1356 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/10164630\/ParabolicWaterTrajectory-225x300-1.jpg\" alt=\"\" width=\"225\" height=\"300\" \/> Fountain spraying in parabolic manner[\/caption]\r\n\r\nParabolic motion and it\u2019s related equations allow us to launch satellites for telecommunications, and rockets for space exploration. Recently, police departments have even begun using projectiles with GPS to track fleeing suspects in vehicles, rather than pursuing them by high-speed chase.\r\n\r\n<section class=\"textbox example\">A small toy rocket is launched from a [latex]4[\/latex]-foot pedestal. The height ([latex]h[\/latex], in feet) of the rocket [latex]t[\/latex] seconds after taking off is given by the formula [latex]h = -2t^2+7t+4[\/latex].\r\n<ol>\r\n \t<li>How long will it take the rocket to hit the ground?\r\n[reveal-answer q=\"529657\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"529657\"]<strong>Read and understand:\u00a0<\/strong>The rocket will be on the ground when the height is [latex]0[\/latex]. We want to know how long, [latex]t[\/latex], \u00a0the rocket is in the air.\r\nLet's substitute [latex]0 [\/latex]for [latex]h[\/latex] and solve for [latex]t[\/latex].\r\n<center>[latex]\\begin{align*} \\text{Given the equation for height} &amp; : &amp; h = -2t^2 + 7t + 4 \\\\ \\text{To find when } h = 0\\text{, factor the quadratic equation} &amp; : &amp; -2t^2 + 7t + 4 = 0 \\\\ \\text{Find two numbers that multiply to } ac = -8 \\text{ and add to } b = 7 &amp; : &amp; \\text{Numbers are } 8 \\text{ and } -1 \\\\ \\text{Rewrite the quadratic using these numbers for factoring} &amp; : &amp; -2t^2 + 8t - t + 4 \\\\ \\text{Group terms to factor by grouping} &amp; : &amp; -2t(t - 4) - 1(t - 4) \\\\ \\text{Factor out the common binomial} &amp; : &amp; (-2t - 1)(t - 4) = 0 \\\\ \\text{Set each factor to zero and solve} &amp; : &amp; \\\\ -2t - 1 = 0 &amp; : &amp; -2t = 1 \\Rightarrow t = -\\frac{1}{2} \\\\ t - 4 = 0 &amp; : &amp; t = 4 \\end{align*}[\/latex]<\/center>Because time cannot be negative, [latex]t = \\frac{1}{2}[\/latex] is not feasible.\r\nThus, the rocket hits the ground at [latex]4[\/latex] seconds.[\/hidden-answer]<\/li>\r\n \t<li>Use the formula for the height of the rocket in the previous example to find the time when the rocket is [latex]4[\/latex] feet from hitting the ground on it\u2019s way back down.\r\n[reveal-answer q=\"437298\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"437298\"]<strong>Read and understand:<\/strong> We are given that the height of the rocket is [latex]4[\/latex] feet from the ground on it\u2019s way back down. We want to know how long it has taken the rocket to get to that point in it\u2019s path, we are going to solve for [latex]t[\/latex].\r\n\r\n[caption id=\"attachment_1357\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1357 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/10165549\/Screen-Shot-2016-06-14-at-5.39.53-PM-300x247-1.png\" alt=\"\" width=\"300\" height=\"247\" \/> Visual of rocket motion[\/caption]\r\n\r\n<center>[latex]\\begin{align*} \\text{Given the height equation:} &amp; \\quad h = -2t^2 + 7t + 4 \\\\ \\text{Set } h = 4 \\text{ feet to find the time:} &amp; \\quad 4 = -2t^2 + 7t + 4 \\\\ \\text{Simplify the equation:} &amp; \\quad 0 = -2t^2 + 7t \\\\ \\text{Factor out the common term } t: &amp; \\quad t(-2t + 7) = 0 \\\\ \\text{Solve each factor for } t: &amp; \\\\ t = 0 &amp; \\quad \\text{(at launch, not relevant for this query)} \\\\ -2t + 7 = 0 &amp; \\quad \\text{Solve: } -2t = -7 \\implies t = \\frac{7}{2} = 3.5 \\text{ seconds} \\\\ \\text{Conclusion:} &amp; \\quad \\text{The rocket is 4 feet above the ground at } t = 3.5 \\text{ seconds.} \\end{align*}[\/latex]<\/center>\r\n[\/hidden-answer]<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]18985[\/ohm2_question]<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]18986[\/ohm2_question]<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]18987[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Construct and apply non-linear equations and formulas to solve real-world problems.<\/li>\n<\/ul>\n<\/section>\n<p>In many real-world situations, relationships between variables are not always straightforward or linear. Non-linear equations and formulas play a crucial role in accurately modeling and solving complex problems across various fields, including physics, engineering, economics, and environmental science.<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Non-linear equations are essential tools in our mathematical toolkit for several reasons:<\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\"><strong>Accurate Representation<\/strong>: Non-linear equations can capture the complexity of real-world phenomena more accurately than linear approximations. Many natural and man-made systems exhibit non-linear behavior, and using these equations allows us to model them faithfully.<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Diverse Applications<\/strong>: The applications of non-linear equations span numerous disciplines:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Physics: Describing planetary motion<\/li>\n<li class=\"whitespace-normal break-words\">Biology: Modeling population growth<\/li>\n<li class=\"whitespace-normal break-words\">Economics: Analyzing market trends<\/li>\n<li class=\"whitespace-normal break-words\">Environmental Science: Predicting climate patterns<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Problem-Solving Power<\/strong>: By constructing and applying non-linear equations, we can:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Solve intricate problems<\/li>\n<li class=\"whitespace-normal break-words\">Make predictions about complex system behavior<\/li>\n<li class=\"whitespace-normal break-words\">Gain insights that might be impossible with simpler linear models<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h2>Quadratic Applications<\/h2>\n<p>Projectile motion happens when you throw a ball into the air and it comes back down because of gravity. \u00a0A projectile will follow a curved path that behaves in a predictable way. \u00a0This predictable motion has been studied for centuries, and in simple cases it\u2019s height from the ground\u00a0at a given time, [latex]t[\/latex], can be modeled with a quadratic polynomial of the form [latex]\\text{height} = at^2+bt+c[\/latex]. Projectile motion is also called a parabolic trajectory because of the shape of the path of a projectile\u2019s motion, as in the image of water in the fountain below.<\/p>\n<figure id=\"attachment_1356\" aria-describedby=\"caption-attachment-1356\" style=\"width: 225px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1356 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/10164630\/ParabolicWaterTrajectory-225x300-1.jpg\" alt=\"\" width=\"225\" height=\"300\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/10164630\/ParabolicWaterTrajectory-225x300-1.jpg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/10164630\/ParabolicWaterTrajectory-225x300-1-65x87.jpg 65w\" sizes=\"(max-width: 225px) 100vw, 225px\" \/><figcaption id=\"caption-attachment-1356\" class=\"wp-caption-text\">Fountain spraying in parabolic manner<\/figcaption><\/figure>\n<p>Parabolic motion and it\u2019s related equations allow us to launch satellites for telecommunications, and rockets for space exploration. Recently, police departments have even begun using projectiles with GPS to track fleeing suspects in vehicles, rather than pursuing them by high-speed chase.<\/p>\n<section class=\"textbox example\">A small toy rocket is launched from a [latex]4[\/latex]-foot pedestal. The height ([latex]h[\/latex], in feet) of the rocket [latex]t[\/latex] seconds after taking off is given by the formula [latex]h = -2t^2+7t+4[\/latex].<\/p>\n<ol>\n<li>How long will it take the rocket to hit the ground?\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q529657\">Show Answer<\/button><\/p>\n<div id=\"q529657\" class=\"hidden-answer\" style=\"display: none\"><strong>Read and understand:\u00a0<\/strong>The rocket will be on the ground when the height is [latex]0[\/latex]. We want to know how long, [latex]t[\/latex], \u00a0the rocket is in the air.<br \/>\nLet&#8217;s substitute [latex]0[\/latex]for [latex]h[\/latex] and solve for [latex]t[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align*} \\text{Given the equation for height} & : & h = -2t^2 + 7t + 4 \\\\ \\text{To find when } h = 0\\text{, factor the quadratic equation} & : & -2t^2 + 7t + 4 = 0 \\\\ \\text{Find two numbers that multiply to } ac = -8 \\text{ and add to } b = 7 & : & \\text{Numbers are } 8 \\text{ and } -1 \\\\ \\text{Rewrite the quadratic using these numbers for factoring} & : & -2t^2 + 8t - t + 4 \\\\ \\text{Group terms to factor by grouping} & : & -2t(t - 4) - 1(t - 4) \\\\ \\text{Factor out the common binomial} & : & (-2t - 1)(t - 4) = 0 \\\\ \\text{Set each factor to zero and solve} & : & \\\\ -2t - 1 = 0 & : & -2t = 1 \\Rightarrow t = -\\frac{1}{2} \\\\ t - 4 = 0 & : & t = 4 \\end{align*}[\/latex]<\/div>\n<p>Because time cannot be negative, [latex]t = \\frac{1}{2}[\/latex] is not feasible.<br \/>\nThus, the rocket hits the ground at [latex]4[\/latex] seconds.<\/p><\/div>\n<\/div>\n<\/li>\n<li>Use the formula for the height of the rocket in the previous example to find the time when the rocket is [latex]4[\/latex] feet from hitting the ground on it\u2019s way back down.\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q437298\">Show Answer<\/button><\/p>\n<div id=\"q437298\" class=\"hidden-answer\" style=\"display: none\"><strong>Read and understand:<\/strong> We are given that the height of the rocket is [latex]4[\/latex] feet from the ground on it\u2019s way back down. We want to know how long it has taken the rocket to get to that point in it\u2019s path, we are going to solve for [latex]t[\/latex].<\/p>\n<figure id=\"attachment_1357\" aria-describedby=\"caption-attachment-1357\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1357 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/10165549\/Screen-Shot-2016-06-14-at-5.39.53-PM-300x247-1.png\" alt=\"\" width=\"300\" height=\"247\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/10165549\/Screen-Shot-2016-06-14-at-5.39.53-PM-300x247-1.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/10165549\/Screen-Shot-2016-06-14-at-5.39.53-PM-300x247-1-65x54.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/10165549\/Screen-Shot-2016-06-14-at-5.39.53-PM-300x247-1-225x185.png 225w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-1357\" class=\"wp-caption-text\">Visual of rocket motion<\/figcaption><\/figure>\n<div style=\"text-align: center;\">[latex]\\begin{align*} \\text{Given the height equation:} & \\quad h = -2t^2 + 7t + 4 \\\\ \\text{Set } h = 4 \\text{ feet to find the time:} & \\quad 4 = -2t^2 + 7t + 4 \\\\ \\text{Simplify the equation:} & \\quad 0 = -2t^2 + 7t \\\\ \\text{Factor out the common term } t: & \\quad t(-2t + 7) = 0 \\\\ \\text{Solve each factor for } t: & \\\\ t = 0 & \\quad \\text{(at launch, not relevant for this query)} \\\\ -2t + 7 = 0 & \\quad \\text{Solve: } -2t = -7 \\implies t = \\frac{7}{2} = 3.5 \\text{ seconds} \\\\ \\text{Conclusion:} & \\quad \\text{The rocket is 4 feet above the ground at } t = 3.5 \\text{ seconds.} \\end{align*}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm18985\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=18985&theme=lumen&iframe_resize_id=ohm18985&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm18986\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=18986&theme=lumen&iframe_resize_id=ohm18986&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm18987\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=18987&theme=lumen&iframe_resize_id=ohm18987&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":12,"menu_order":19,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":92,"module-header":"learn_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\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1355"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/12"}],"version-history":[{"count":11,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1355\/revisions"}],"predecessor-version":[{"id":7616,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1355\/revisions\/7616"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/92"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1355\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=1355"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1355"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=1355"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=1355"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}