{"id":1993,"date":"2021-08-19T16:07:01","date_gmt":"2021-08-19T16:07:01","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/skills-review-for-the-logistic-equation\/"},"modified":"2021-11-19T03:12:34","modified_gmt":"2021-11-19T03:12:34","slug":"skills-review-for-the-logistic-equation","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/skills-review-for-the-logistic-equation\/","title":{"raw":"Skills Review for The Logistic Equation","rendered":"Skills Review for The Logistic Equation"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcome<\/h3>\r\n<ul>\r\n \t<li>Apply continuous growth\/decay models<\/li>\r\n<\/ul>\r\n<\/div>\r\nThe Logistic Equation section will expose us to differential equations and population growth and carrying capacity. Here we will review exponential models.\r\n<h2>Use Exponential Models<\/h2>\r\n<p id=\"fs-id1165137827929\">For most real-world phenomena, however, <em>e <\/em>is used as the base for exponential functions. Exponential models that use <em>e<\/em>\u00a0as the base are called <em>continuous growth or decay models<\/em>. We see these models in finance, computer science, and most of the sciences, such as physics, toxicology, and fluid dynamics.<\/p>\r\n\r\n<div id=\"fs-id1165137664673\" class=\"note textbox\">\r\n<h3 class=\"title\">The Continuous Growth\/Decay Formula<\/h3>\r\n<p id=\"fs-id1165135453868\">For all real numbers r,\u00a0<em>t<\/em>, and all positive numbers <em>a<\/em>, continuous growth or decay is represented by the formula<\/p>\r\n\r\n<div id=\"fs-id1165135536370\" class=\"equation\" style=\"text-align: center;\">[latex]P\\left(t\\right)=P_0{e}^{rt}[\/latex]<\/div>\r\n<p id=\"eip-101\">where<\/p>\r\n\r\n<ul id=\"fs-id1165135152052\">\r\n \t<li>[latex]P_0[\/latex]\u00a0is the initial value,<\/li>\r\n \t<li><em>r<\/em>\u00a0is the continuous growth or decay rate per unit time,<\/li>\r\n \t<li>and <em>t<\/em>\u00a0is the elapsed time.<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165135560686\">If <em>r\u00a0<\/em>&gt;[latex]0[\/latex], then the formula represents continuous growth. If <em>r\u00a0<\/em>&lt;\u00a0[latex]0[\/latex], then the formula represents continuous decay.<\/p>\r\n<p id=\"fs-id1165137812323\">For business applications, the continuous growth formula is called the continuous compounding formula and takes the form<\/p>\r\n\r\n<div id=\"eip-id1165134324899\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]A\\left(t\\right)=P_0{e}^{rt}[\/latex]<\/div>\r\n<p id=\"eip-962\">where<\/p>\r\n\r\n<ul id=\"fs-id1165137827330\">\r\n \t<li>[latex]P_0[\/latex]\u00a0is the principal or the initial amount invested,<\/li>\r\n \t<li><em>r<\/em>\u00a0is the growth or interest rate per unit time,<\/li>\r\n \t<li>and <em>t<\/em>\u00a0is the period or term of the investment.<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn our next example, we will calculate continuous growth of an investment. It is important to note the language that is used in the instructions for interest rate problems. \u00a0You will know to use the <em>continuous<\/em> growth or decay formula when you are asked to find an amount based on continuous compounding.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using the Continuously Compounded Interest Formula<\/h3>\r\nA person invested\u00a0[latex]$1,000[\/latex] in an account earning a nominal\u00a0[latex]10\\%[\/latex] per year compounded continuously. How much was in the account at the end of one year?\r\n[reveal-answer q=\"33008\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"33008\"]\r\n\r\nSince the account is growing in value, this is a continuous compounding problem with growth rate <em>r\u00a0<\/em>=[latex]0.10[\/latex]. The initial investment was\u00a0[latex]$1,000[\/latex], so <em>P\u00a0<\/em>=[latex]1000[\/latex]. We use the continuous compounding formula to find the value after <em>t\u00a0<\/em>=[latex]1[\/latex] year:\r\n<div id=\"eip-id1165133351794\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{c}A\\left(t\\right)\\hfill &amp; =P{e}^{rt}\\hfill &amp; \\text{Use the continuous compounding formula}.\\hfill \\\\ \\hfill &amp; =1000{\\left(e\\right)}^{0.1} &amp; \\text{Substitute known values for }P, r,\\text{ and }t.\\hfill \\\\ \\hfill &amp; \\approx 1105.17\\hfill &amp; \\text{Use a calculator to approximate}.\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1165137895288\">The account is worth\u00a0[latex]$1,105.17[\/latex] after one year.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135411368\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165135411373\">How To: Given the initial value, rate of growth or decay, and time [latex]t[\/latex], solve a continuous growth or decay function<\/h3>\r\n<ol id=\"fs-id1165135511371\">\r\n \t<li>Use the information in the problem to determine\u00a0[latex]P_0[\/latex], the initial value of the function.<\/li>\r\n \t<li>Use the information in the problem to determine the growth rate <em>r<\/em>.\r\n<ol id=\"fs-id1165135188096\">\r\n \t<li>If the problem refers to continuous growth, then <em>r\u00a0<\/em>&gt; [latex]0[\/latex].<\/li>\r\n \t<li>If the problem refers to continuous decay, then <em>r\u00a0<\/em>&lt;\u00a0[latex]0[\/latex].<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Use the information in the problem to determine the time <em>t<\/em>.<\/li>\r\n \t<li>Substitute the given information into the continuous growth formula and solve for P(<em>t<\/em>).<\/li>\r\n<\/ol>\r\n<\/div>\r\nIn our next example, we will calculate continuous decay. Pay attention to the rate - it is negative which means we are considering a situation where an amount decreases or decays.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using the Continuous Decay Model<\/h3>\r\nRadon-222 decays at a continuous rate of\u00a0[latex]17.3\\%[\/latex] per day. How much will\u00a0[latex]100[\/latex] mg of Radon-[latex]222[\/latex] decay to in\u00a0[latex]3[\/latex] days?\r\n\r\n[reveal-answer q=\"995802\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"995802\"]\r\n\r\nSince the substance is decaying, the rate,\u00a0[latex]17.3\\%[\/latex], is negative. So, <em>r\u00a0<\/em>=\u00a0[latex]\u20130.173[\/latex]. The initial amount of radon-[latex]222[\/latex] was [latex]100[\/latex] mg, so [latex]P_0=100[\/latex]. We use the continuous decay formula to find the value after <em>t\u00a0<\/em>=[latex]3[\/latex] days:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}P\\left(t\\right)\\hfill &amp; =P_0{e}^{rt}\\hfill &amp; \\text{Use the continuous growth formula}.\\hfill \\\\ \\hfill &amp; =100{e}^{-0.173\\left(3\\right)} &amp; \\text{Substitute known values for }a, r,\\text{ and }t.\\hfill \\\\ \\hfill &amp; \\approx 59.5115\\hfill &amp; \\text{Use a calculator to approximate}.\\hfill \\end{array}[\/latex]<\/p>\r\n<p id=\"fs-id1165137697132\">So\u00a0[latex]59.5115[\/latex] mg of radon-[latex]222[\/latex] will remain.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nA certain bacteria has an initial population of [latex]1000[\/latex] with a growth rate of 3%. What will the population of the bacteria be in 5 years?\r\n\r\n[reveal-answer q=\"9958021\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"9958021\"]\r\n\r\n<span style=\"font-size: 1rem; text-align: initial;\">The population will be [latex]1162[\/latex].<\/span>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcome<\/h3>\n<ul>\n<li>Apply continuous growth\/decay models<\/li>\n<\/ul>\n<\/div>\n<p>The Logistic Equation section will expose us to differential equations and population growth and carrying capacity. Here we will review exponential models.<\/p>\n<h2>Use Exponential Models<\/h2>\n<p id=\"fs-id1165137827929\">For most real-world phenomena, however, <em>e <\/em>is used as the base for exponential functions. Exponential models that use <em>e<\/em>\u00a0as the base are called <em>continuous growth or decay models<\/em>. We see these models in finance, computer science, and most of the sciences, such as physics, toxicology, and fluid dynamics.<\/p>\n<div id=\"fs-id1165137664673\" class=\"note textbox\">\n<h3 class=\"title\">The Continuous Growth\/Decay Formula<\/h3>\n<p id=\"fs-id1165135453868\">For all real numbers r,\u00a0<em>t<\/em>, and all positive numbers <em>a<\/em>, continuous growth or decay is represented by the formula<\/p>\n<div id=\"fs-id1165135536370\" class=\"equation\" style=\"text-align: center;\">[latex]P\\left(t\\right)=P_0{e}^{rt}[\/latex]<\/div>\n<p id=\"eip-101\">where<\/p>\n<ul id=\"fs-id1165135152052\">\n<li>[latex]P_0[\/latex]\u00a0is the initial value,<\/li>\n<li><em>r<\/em>\u00a0is the continuous growth or decay rate per unit time,<\/li>\n<li>and <em>t<\/em>\u00a0is the elapsed time.<\/li>\n<\/ul>\n<p id=\"fs-id1165135560686\">If <em>r\u00a0<\/em>&gt;[latex]0[\/latex], then the formula represents continuous growth. If <em>r\u00a0<\/em>&lt;\u00a0[latex]0[\/latex], then the formula represents continuous decay.<\/p>\n<p id=\"fs-id1165137812323\">For business applications, the continuous growth formula is called the continuous compounding formula and takes the form<\/p>\n<div id=\"eip-id1165134324899\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]A\\left(t\\right)=P_0{e}^{rt}[\/latex]<\/div>\n<p id=\"eip-962\">where<\/p>\n<ul id=\"fs-id1165137827330\">\n<li>[latex]P_0[\/latex]\u00a0is the principal or the initial amount invested,<\/li>\n<li><em>r<\/em>\u00a0is the growth or interest rate per unit time,<\/li>\n<li>and <em>t<\/em>\u00a0is the period or term of the investment.<\/li>\n<\/ul>\n<\/div>\n<p>In our next example, we will calculate continuous growth of an investment. It is important to note the language that is used in the instructions for interest rate problems. \u00a0You will know to use the <em>continuous<\/em> growth or decay formula when you are asked to find an amount based on continuous compounding.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Continuously Compounded Interest Formula<\/h3>\n<p>A person invested\u00a0[latex]$1,000[\/latex] in an account earning a nominal\u00a0[latex]10\\%[\/latex] per year compounded continuously. How much was in the account at the end of one year?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q33008\">Show Solution<\/span><\/p>\n<div id=\"q33008\" class=\"hidden-answer\" style=\"display: none\">\n<p>Since the account is growing in value, this is a continuous compounding problem with growth rate <em>r\u00a0<\/em>=[latex]0.10[\/latex]. The initial investment was\u00a0[latex]$1,000[\/latex], so <em>P\u00a0<\/em>=[latex]1000[\/latex]. We use the continuous compounding formula to find the value after <em>t\u00a0<\/em>=[latex]1[\/latex] year:<\/p>\n<div id=\"eip-id1165133351794\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{c}A\\left(t\\right)\\hfill & =P{e}^{rt}\\hfill & \\text{Use the continuous compounding formula}.\\hfill \\\\ \\hfill & =1000{\\left(e\\right)}^{0.1} & \\text{Substitute known values for }P, r,\\text{ and }t.\\hfill \\\\ \\hfill & \\approx 1105.17\\hfill & \\text{Use a calculator to approximate}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137895288\">The account is worth\u00a0[latex]$1,105.17[\/latex] after one year.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135411368\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165135411373\">How To: Given the initial value, rate of growth or decay, and time [latex]t[\/latex], solve a continuous growth or decay function<\/h3>\n<ol id=\"fs-id1165135511371\">\n<li>Use the information in the problem to determine\u00a0[latex]P_0[\/latex], the initial value of the function.<\/li>\n<li>Use the information in the problem to determine the growth rate <em>r<\/em>.\n<ol id=\"fs-id1165135188096\">\n<li>If the problem refers to continuous growth, then <em>r\u00a0<\/em>&gt; [latex]0[\/latex].<\/li>\n<li>If the problem refers to continuous decay, then <em>r\u00a0<\/em>&lt;\u00a0[latex]0[\/latex].<\/li>\n<\/ol>\n<\/li>\n<li>Use the information in the problem to determine the time <em>t<\/em>.<\/li>\n<li>Substitute the given information into the continuous growth formula and solve for P(<em>t<\/em>).<\/li>\n<\/ol>\n<\/div>\n<p>In our next example, we will calculate continuous decay. Pay attention to the rate &#8211; it is negative which means we are considering a situation where an amount decreases or decays.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Continuous Decay Model<\/h3>\n<p>Radon-222 decays at a continuous rate of\u00a0[latex]17.3\\%[\/latex] per day. How much will\u00a0[latex]100[\/latex] mg of Radon-[latex]222[\/latex] decay to in\u00a0[latex]3[\/latex] days?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q995802\">Show Solution<\/span><\/p>\n<div id=\"q995802\" class=\"hidden-answer\" style=\"display: none\">\n<p>Since the substance is decaying, the rate,\u00a0[latex]17.3\\%[\/latex], is negative. So, <em>r\u00a0<\/em>=\u00a0[latex]\u20130.173[\/latex]. The initial amount of radon-[latex]222[\/latex] was [latex]100[\/latex] mg, so [latex]P_0=100[\/latex]. We use the continuous decay formula to find the value after <em>t\u00a0<\/em>=[latex]3[\/latex] days:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}P\\left(t\\right)\\hfill & =P_0{e}^{rt}\\hfill & \\text{Use the continuous growth formula}.\\hfill \\\\ \\hfill & =100{e}^{-0.173\\left(3\\right)} & \\text{Substitute known values for }a, r,\\text{ and }t.\\hfill \\\\ \\hfill & \\approx 59.5115\\hfill & \\text{Use a calculator to approximate}.\\hfill \\end{array}[\/latex]<\/p>\n<p id=\"fs-id1165137697132\">So\u00a0[latex]59.5115[\/latex] mg of radon-[latex]222[\/latex] will remain.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>A certain bacteria has an initial population of [latex]1000[\/latex] with a growth rate of 3%. What will the population of the bacteria be in 5 years?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q9958021\">Show Solution<\/span><\/p>\n<div id=\"q9958021\" class=\"hidden-answer\" style=\"display: none\">\n<p><span style=\"font-size: 1rem; text-align: initial;\">The population will be [latex]1162[\/latex].<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1993\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Modification and Revision. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra Corequisite. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/\">https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Precalculus. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/precalculus\/\">https:\/\/courses.lumenlearning.com\/precalculus\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":349141,"menu_order":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Modification and Revision\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra Corequisite\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"https:\/\/courses.lumenlearning.com\/precalculus\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1993","chapter","type-chapter","status-publish","hentry"],"part":536,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1993","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/349141"}],"version-history":[{"count":5,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1993\/revisions"}],"predecessor-version":[{"id":2556,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1993\/revisions\/2556"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/536"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1993\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=1993"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=1993"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=1993"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=1993"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}