{"id":2377,"date":"2024-07-25T00:44:03","date_gmt":"2024-07-25T00:44:03","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=2377"},"modified":"2024-12-16T20:24:39","modified_gmt":"2024-12-16T20:24:39","slug":"systems-of-nonlinear-equations-and-inequalities-apply-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/systems-of-nonlinear-equations-and-inequalities-apply-it-1\/","title":{"raw":"Systems of Nonlinear Equations and Inequalities: Apply It 1","rendered":"Systems of Nonlinear Equations and Inequalities: Apply It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Solve equations with squared variables or other exponents using substitution and elimination<\/li>\r\n \t<li>Graph curved inequalities and find where they overlap<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Real-World Applications<\/h2>\r\nReady to tackle a real-world challenge using your math skills? Imagine you're solving a practical problem that involves finding the dimensions or quantities for various objects under specific constraints. Let's dive into how you can use nonlinear equations to find the solution!\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">You're tasked with designing an inclusive community center that features a rectangular activity room and a circular meditation space. The center aims to be accessible and welcoming, providing a variety of activities and quiet spaces for reflection and relaxation.\r\n<ul>\r\n \t<li>The activity room needs to be large enough to host diverse group activities and should have an area of [latex]200[\/latex] square meters.<\/li>\r\n \t<li>The length of the activity room is [latex]5[\/latex] meters more than its width.<\/li>\r\n \t<li>The meditation space should have an area of [latex]50[\/latex] square meters, providing a peaceful environment for mindfulness and relaxation.<\/li>\r\n<\/ul>\r\nDetermine the optimal dimensions for these spaces while adhering to the stated requirements.\r\n\r\n[reveal-answer q=\"79899\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"79899\"]\r\n<p class=\"whitespace-pre-wrap break-words\">Let's solve for the rectangular activity room first<\/p>\r\n\r\n<ul class=\"-mt-1 [li&gt;&amp;]:mt-2 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Let's call the width of the room [latex]w[\/latex] meters<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Given that length is [latex]5[\/latex] meters more than width, length = [latex]w + 5[\/latex] meters<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Area of a rectangle = length [latex]\\times[\/latex] width<\/li>\r\n \t<li class=\"whitespace-normal break-words\">We know area = [latex]200 m^2[\/latex]<\/li>\r\n<\/ul>\r\nSet up and solve the equation:\r\n\r\n<center>[latex]\r\n\\begin{array}{rcl}\r\n200 &amp;=&amp; w(w + 5) \\\\\r\n200 &amp;=&amp; w^2 + 5 \\times w \\\\\r\n0 &amp;=&amp; w^2 + 5w - 200\r\n\\end{array}\r\n[\/latex]<\/center>\r\nUsing quadratic formula:\r\n\r\n<center>[latex]\r\n\\begin{array}{rcl}\r\nw &amp;=&amp; \\frac{-5 \\pm \\sqrt{25 + 800}}{2} \\\\\r\nw &amp;=&amp; \\frac{-5 \\pm \\sqrt{825}}{2} \\\\\r\nw &amp;=&amp; \\frac{-5 \\pm 28.7228}{2} \\\\\r\nw &amp;=&amp; 11.8614 \\text{ or } -16.8614\r\n\\end{array}\r\n[\/latex]<\/center>Since width can't be negative, [latex]w \\approx 11.86[\/latex] meters\r\nTherefore, length = [latex]11.86 + 5 = 16.86[\/latex] meters\r\n\r\nFor the circular meditation space, the area of a circle = [latex]\\pi r^2[\/latex].\r\n\r\n<center>[latex]\r\n\\begin{array}{rcl}\r\n50 &amp;=&amp; \\pi r^2 \\\\\r\nr^2 &amp;=&amp; \\frac{50}{\\pi} \\\\\r\nr &amp;=&amp; \\sqrt{\\frac{50}{\\pi}} \\\\\r\nr &amp;\\approx&amp; 3.99 \\text{ meters}\r\n\\end{array}\r\n[\/latex]<\/center>\r\n<p class=\"whitespace-pre-wrap break-words\">Therefore, the dimensions are:<\/p>\r\nActivity Room:\r\n<ul class=\"-mt-1 [li&gt;&amp;]:mt-2 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Width [latex]\\approx 11.86[\/latex] meters<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Length [latex]\\approx 16.86[\/latex] meters<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Area = [latex]200[\/latex] square meters<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-pre-wrap break-words\">Meditation Space:<\/p>\r\n\r\n<ul class=\"-mt-1 [li&gt;&amp;]:mt-2 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Radius [latex]\\approx 3.99[\/latex] meters<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Area = [latex]50[\/latex] square meters<\/li>\r\n<\/ul>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm2_question hide_question_numbers=1]24822[\/ohm2_question]<\/section><section aria-label=\"Try It\"><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm2_question hide_question_numbers=1]31662[\/ohm2_question]<\/section><section aria-label=\"Try It\"><\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm2_question hide_question_numbers=1]31663[\/ohm2_question]<\/section><section aria-label=\"Try It\"><\/section><\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Solve equations with squared variables or other exponents using substitution and elimination<\/li>\n<li>Graph curved inequalities and find where they overlap<\/li>\n<\/ul>\n<\/section>\n<h2>Real-World Applications<\/h2>\n<p>Ready to tackle a real-world challenge using your math skills? Imagine you&#8217;re solving a practical problem that involves finding the dimensions or quantities for various objects under specific constraints. Let&#8217;s dive into how you can use nonlinear equations to find the solution!<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">You&#8217;re tasked with designing an inclusive community center that features a rectangular activity room and a circular meditation space. The center aims to be accessible and welcoming, providing a variety of activities and quiet spaces for reflection and relaxation.<\/p>\n<ul>\n<li>The activity room needs to be large enough to host diverse group activities and should have an area of [latex]200[\/latex] square meters.<\/li>\n<li>The length of the activity room is [latex]5[\/latex] meters more than its width.<\/li>\n<li>The meditation space should have an area of [latex]50[\/latex] square meters, providing a peaceful environment for mindfulness and relaxation.<\/li>\n<\/ul>\n<p>Determine the optimal dimensions for these spaces while adhering to the stated requirements.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q79899\">Show Answer<\/button><\/p>\n<div id=\"q79899\" class=\"hidden-answer\" style=\"display: none\">\n<p class=\"whitespace-pre-wrap break-words\">Let&#8217;s solve for the rectangular activity room first<\/p>\n<ul class=\"-mt-1 &#091;li&gt;&amp;&#093;:mt-2 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Let&#8217;s call the width of the room [latex]w[\/latex] meters<\/li>\n<li class=\"whitespace-normal break-words\">Given that length is [latex]5[\/latex] meters more than width, length = [latex]w + 5[\/latex] meters<\/li>\n<li class=\"whitespace-normal break-words\">Area of a rectangle = length [latex]\\times[\/latex] width<\/li>\n<li class=\"whitespace-normal break-words\">We know area = [latex]200 m^2[\/latex]<\/li>\n<\/ul>\n<p>Set up and solve the equation:<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{rcl}  200 &=& w(w + 5) \\\\  200 &=& w^2 + 5 \\times w \\\\  0 &=& w^2 + 5w - 200  \\end{array}[\/latex]<\/div>\n<p>Using quadratic formula:<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{rcl}  w &=& \\frac{-5 \\pm \\sqrt{25 + 800}}{2} \\\\  w &=& \\frac{-5 \\pm \\sqrt{825}}{2} \\\\  w &=& \\frac{-5 \\pm 28.7228}{2} \\\\  w &=& 11.8614 \\text{ or } -16.8614  \\end{array}[\/latex]<\/div>\n<p>Since width can&#8217;t be negative, [latex]w \\approx 11.86[\/latex] meters<br \/>\nTherefore, length = [latex]11.86 + 5 = 16.86[\/latex] meters<\/p>\n<p>For the circular meditation space, the area of a circle = [latex]\\pi r^2[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{rcl}  50 &=& \\pi r^2 \\\\  r^2 &=& \\frac{50}{\\pi} \\\\  r &=& \\sqrt{\\frac{50}{\\pi}} \\\\  r &\\approx& 3.99 \\text{ meters}  \\end{array}[\/latex]<\/div>\n<p class=\"whitespace-pre-wrap break-words\">Therefore, the dimensions are:<\/p>\n<p>Activity Room:<\/p>\n<ul class=\"-mt-1 &#091;li&gt;&amp;&#093;:mt-2 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Width [latex]\\approx 11.86[\/latex] meters<\/li>\n<li class=\"whitespace-normal break-words\">Length [latex]\\approx 16.86[\/latex] meters<\/li>\n<li class=\"whitespace-normal break-words\">Area = [latex]200[\/latex] square meters<\/li>\n<\/ul>\n<p class=\"whitespace-pre-wrap break-words\">Meditation Space:<\/p>\n<ul class=\"-mt-1 &#091;li&gt;&amp;&#093;:mt-2 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Radius [latex]\\approx 3.99[\/latex] meters<\/li>\n<li class=\"whitespace-normal break-words\">Area = [latex]50[\/latex] square meters<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm24822\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=24822&theme=lumen&iframe_resize_id=ohm24822&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section aria-label=\"Try It\">\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm31662\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=31662&theme=lumen&iframe_resize_id=ohm31662&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section aria-label=\"Try It\"><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm31663\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=31663&theme=lumen&iframe_resize_id=ohm31663&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section aria-label=\"Try It\"><\/section>\n<\/section>\n","protected":false},"author":12,"menu_order":23,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":300,"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\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2377"}],"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":14,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2377\/revisions"}],"predecessor-version":[{"id":6760,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2377\/revisions\/6760"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/300"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2377\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=2377"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2377"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=2377"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=2377"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}