{"id":1387,"date":"2024-05-10T21:56:01","date_gmt":"2024-05-10T21:56:01","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=1387"},"modified":"2025-01-16T21:42:17","modified_gmt":"2025-01-16T21:42:17","slug":"applications-and-inequalities-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/applications-and-inequalities-fresh-take\/","title":{"raw":"Applications of Non-Linear Equations: Fresh Take","rendered":"Applications of Non-Linear Equations: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li>Build and use equations and formulas that aren't straight lines to solve real-life problems.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Quadratic Applications<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Projectile Motion: The curved path of an object launched into the air and affected by gravity<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Quadratic Model: [latex]h = at^2 + bt + c[\/latex], where:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]h[\/latex] is height<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]t[\/latex] is time<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex] are constants<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Parabolic Trajectory: The shape of the path followed by a projectile<\/li>\r\n \t<li>Real-world appliations\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Satellite launches<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Rocket science<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Sports (e.g., basketball shots, football passes)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Ballistics<\/li>\r\n \t<li class=\"whitespace-normal break-words\">GPS-enabled tracking projectiles in law enforcement<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Problem-Solving Strategy<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify the given quadratic equation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Determine what information is required (e.g., time to hit ground, maximum height)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Set up the equation based on the question:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">For ground impact: Set height to 0<\/li>\r\n \t<li class=\"whitespace-normal break-words\">For specific height: Set height to the given value<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve the resulting quadratic equation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Interpret the results in the context of the problem<\/li>\r\n<\/ol>\r\n<strong>\u00a0<\/strong>\r\n\r\n<\/div>\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bbacccae-4-f9KxGRXnU\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/4-f9KxGRXnU?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-bbacccae-4-f9KxGRXnU\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12844264&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-bbacccae-4-f9KxGRXnU&vembed=0&video_id=4-f9KxGRXnU&video_target=tpm-plugin-bbacccae-4-f9KxGRXnU'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Quadratic+Function+Application+Using+Formulas+-+Rocket+Launch_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Quadratic Function Application Using Formulas - Rocket Launch\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Areas and Volumes<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Geometric Formulas:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Area formulas (2D shapes)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Volume formulas (3D shapes)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Perimeter and surface area<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Variable Relationships:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Linear (e.g., perimeter of a rectangle)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Quadratic (e.g., area of a circle)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Cubic (e.g., volume of a sphere)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Problem-Solving Approach:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify relevant shapes<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Determine given and unknown variables<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Select appropriate formulas<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Set up and solve equations<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Interpret results in context<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<table style=\"width: 100%; border-collapse: collapse; margin-bottom: 20px;\">\r\n<thead>\r\n<tr style=\"background-color: #f2f2f2;\">\r\n<th style=\"border: 1px solid #ddd; padding: 8px; text-align: left;\">Shape<\/th>\r\n<th style=\"border: 1px solid #ddd; padding: 8px; text-align: left;\">Formula<\/th>\r\n<th style=\"border: 1px solid #ddd; padding: 8px; text-align: left;\">Variables<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"border: 1px solid #ddd; padding: 8px;\">Rectangle (Area)<\/td>\r\n<td style=\"border: 1px solid #ddd; padding: 8px;\">[latex]A = lw[\/latex]<\/td>\r\n<td style=\"border: 1px solid #ddd; padding: 8px;\">[latex]l[\/latex] = length, [latex]w[\/latex] = width<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"border: 1px solid #ddd; padding: 8px;\">Circle (Area)<\/td>\r\n<td style=\"border: 1px solid #ddd; padding: 8px;\">[latex]A = \\pi r^2[\/latex]<\/td>\r\n<td style=\"border: 1px solid #ddd; padding: 8px;\">[latex]r [\/latex] = radius<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"border: 1px solid #ddd; padding: 8px;\">Triangle (Area)<\/td>\r\n<td style=\"border: 1px solid #ddd; padding: 8px;\">[latex]A = \\frac{1}{2}bh[\/latex]<\/td>\r\n<td style=\"border: 1px solid #ddd; padding: 8px;\">[latex]b[\/latex] = base, [latex]h[\/latex] = height<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"border: 1px solid #ddd; padding: 8px;\">Rectangular Prism (Volume)<\/td>\r\n<td style=\"border: 1px solid #ddd; padding: 8px;\">[latex]V = lwh[\/latex]<\/td>\r\n<td style=\"border: 1px solid #ddd; padding: 8px;\">[latex]l[\/latex] = length, [latex]w[\/latex] = width, [latex]h[\/latex] = height<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"border: 1px solid #ddd; padding: 8px;\">Sphere (Volume)<\/td>\r\n<td style=\"border: 1px solid #ddd; padding: 8px;\">[latex]V = \\frac{4}{3}\\pi r^3[\/latex]<\/td>\r\n<td style=\"border: 1px solid #ddd; padding: 8px;\">[latex]r[\/latex] = radius<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Build and use equations and formulas that aren&#8217;t straight lines to solve real-life problems.<\/li>\n<\/ul>\n<\/section>\n<h2>Quadratic Applications<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Projectile Motion: The curved path of an object launched into the air and affected by gravity<\/li>\n<li class=\"whitespace-normal break-words\">Quadratic Model: [latex]h = at^2 + bt + c[\/latex], where:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]h[\/latex] is height<\/li>\n<li class=\"whitespace-normal break-words\">[latex]t[\/latex] is time<\/li>\n<li class=\"whitespace-normal break-words\">[latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex] are constants<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Parabolic Trajectory: The shape of the path followed by a projectile<\/li>\n<li>Real-world appliations\n<ul>\n<li class=\"whitespace-normal break-words\">Satellite launches<\/li>\n<li class=\"whitespace-normal break-words\">Rocket science<\/li>\n<li class=\"whitespace-normal break-words\">Sports (e.g., basketball shots, football passes)<\/li>\n<li class=\"whitespace-normal break-words\">Ballistics<\/li>\n<li class=\"whitespace-normal break-words\">GPS-enabled tracking projectiles in law enforcement<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Problem-Solving Strategy<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify the given quadratic equation<\/li>\n<li class=\"whitespace-normal break-words\">Determine what information is required (e.g., time to hit ground, maximum height)<\/li>\n<li class=\"whitespace-normal break-words\">Set up the equation based on the question:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For ground impact: Set height to 0<\/li>\n<li class=\"whitespace-normal break-words\">For specific height: Set height to the given value<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Solve the resulting quadratic equation<\/li>\n<li class=\"whitespace-normal break-words\">Interpret the results in the context of the problem<\/li>\n<\/ol>\n<p><strong>\u00a0<\/strong><\/p>\n<\/div>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bbacccae-4-f9KxGRXnU\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/4-f9KxGRXnU?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-bbacccae-4-f9KxGRXnU\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12844264&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-bbacccae-4-f9KxGRXnU&#38;vembed=0&#38;video_id=4-f9KxGRXnU&#38;video_target=tpm-plugin-bbacccae-4-f9KxGRXnU\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Quadratic+Function+Application+Using+Formulas+-+Rocket+Launch_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Quadratic Function Application Using Formulas &#8211; Rocket Launch\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Areas and Volumes<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Geometric Formulas:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Area formulas (2D shapes)<\/li>\n<li class=\"whitespace-normal break-words\">Volume formulas (3D shapes)<\/li>\n<li class=\"whitespace-normal break-words\">Perimeter and surface area<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Variable Relationships:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Linear (e.g., perimeter of a rectangle)<\/li>\n<li class=\"whitespace-normal break-words\">Quadratic (e.g., area of a circle)<\/li>\n<li class=\"whitespace-normal break-words\">Cubic (e.g., volume of a sphere)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Problem-Solving Approach:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify relevant shapes<\/li>\n<li class=\"whitespace-normal break-words\">Determine given and unknown variables<\/li>\n<li class=\"whitespace-normal break-words\">Select appropriate formulas<\/li>\n<li class=\"whitespace-normal break-words\">Set up and solve equations<\/li>\n<li class=\"whitespace-normal break-words\">Interpret results in context<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<table style=\"width: 100%; border-collapse: collapse; margin-bottom: 20px;\">\n<thead>\n<tr style=\"background-color: #f2f2f2;\">\n<th style=\"border: 1px solid #ddd; padding: 8px; text-align: left;\">Shape<\/th>\n<th style=\"border: 1px solid #ddd; padding: 8px; text-align: left;\">Formula<\/th>\n<th style=\"border: 1px solid #ddd; padding: 8px; text-align: left;\">Variables<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"border: 1px solid #ddd; padding: 8px;\">Rectangle (Area)<\/td>\n<td style=\"border: 1px solid #ddd; padding: 8px;\">[latex]A = lw[\/latex]<\/td>\n<td style=\"border: 1px solid #ddd; padding: 8px;\">[latex]l[\/latex] = length, [latex]w[\/latex] = width<\/td>\n<\/tr>\n<tr>\n<td style=\"border: 1px solid #ddd; padding: 8px;\">Circle (Area)<\/td>\n<td style=\"border: 1px solid #ddd; padding: 8px;\">[latex]A = \\pi r^2[\/latex]<\/td>\n<td style=\"border: 1px solid #ddd; padding: 8px;\">[latex]r[\/latex] = radius<\/td>\n<\/tr>\n<tr>\n<td style=\"border: 1px solid #ddd; padding: 8px;\">Triangle (Area)<\/td>\n<td style=\"border: 1px solid #ddd; padding: 8px;\">[latex]A = \\frac{1}{2}bh[\/latex]<\/td>\n<td style=\"border: 1px solid #ddd; padding: 8px;\">[latex]b[\/latex] = base, [latex]h[\/latex] = height<\/td>\n<\/tr>\n<tr>\n<td style=\"border: 1px solid #ddd; padding: 8px;\">Rectangular Prism (Volume)<\/td>\n<td style=\"border: 1px solid #ddd; padding: 8px;\">[latex]V = lwh[\/latex]<\/td>\n<td style=\"border: 1px solid #ddd; padding: 8px;\">[latex]l[\/latex] = length, [latex]w[\/latex] = width, [latex]h[\/latex] = height<\/td>\n<\/tr>\n<tr>\n<td style=\"border: 1px solid #ddd; padding: 8px;\">Sphere (Volume)<\/td>\n<td style=\"border: 1px solid #ddd; padding: 8px;\">[latex]V = \\frac{4}{3}\\pi r^3[\/latex]<\/td>\n<td style=\"border: 1px solid #ddd; padding: 8px;\">[latex]r[\/latex] = radius<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n","protected":false},"author":12,"menu_order":23,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Ex: Quadratic Function Application Using Formulas - Rocket Launch \",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/www.youtube.com\/watch?v=4-f9KxGRXnU\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube 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