{"id":1685,"date":"2015-11-12T18:30:46","date_gmt":"2015-11-12T18:30:46","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&#038;p=1685"},"modified":"2017-04-03T17:42:36","modified_gmt":"2017-04-03T17:42:36","slug":"key-concepts-glossary-34","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/chapter\/key-concepts-glossary-34\/","title":{"raw":"Key Concepts &amp; Glossary","rendered":"Key Concepts &amp; Glossary"},"content":{"raw":"<section id=\"fs-id1165135208780\" class=\"key-equations\" data-depth=\"1\"><h2 data-type=\"title\">Key Equations<\/h2>\r\n<table id=\"fs-id2347772\" summary=\"..\"><tbody><tr><td>Half-life formula<\/td>\r\n<td>If [latex]\\text{ }A={A}_{0}{e}^{kt}[\/latex], <em>k\u00a0<\/em>&lt; 0, the half-life is [latex]t=-\\frac{\\mathrm{ln}\\left(2\\right)}{k}[\/latex].<\/td>\r\n<\/tr><tr><td>Carbon-14 dating<\/td>\r\n<td>\n\n\r\n\r\n[latex]t=\\frac{\\mathrm{ln}\\left(\\frac{A}{{A}_{0}}\\right)}{-0.000121}[\/latex].[latex]{A}_{0}[\/latex] <em>A<\/em>\u00a0is the amount of carbon-14 when the plant or animal died\r\n\r\n<em>t<\/em>\u00a0is the amount of carbon-14 remaining today\r\n\r\nis the age of the fossil in years<\/td>\r\n<\/tr><tr><td>Doubling time formula<\/td>\r\n<td>If [latex]A={A}_{0}{e}^{kt}[\/latex], <em>k\u00a0<\/em>&gt; 0, the doubling time is [latex]t=\\frac{\\mathrm{ln}2}{k}[\/latex]<\/td>\r\n<\/tr><tr><td>Newton\u2019s Law of Cooling<\/td>\r\n<td>[latex]T\\left(t\\right)=A{e}^{kt}+{T}_{s}[\/latex], where [latex]{T}_{s}[\/latex] is the ambient temperature, [latex]A=T\\left(0\\right)-{T}_{s}[\/latex], and <em>k<\/em> is the continuous rate of cooling.<\/td>\r\n<\/tr><\/tbody><\/table><\/section><section id=\"fs-id1165137894245\" class=\"key-concepts\" data-depth=\"1\"><h2 data-type=\"title\">Key Concepts<\/h2>\r\n<ul id=\"fs-id1165137894248\"><li>The basic exponential function is [latex]f\\left(x\\right)=a{b}^{x}[\/latex]. If <em>b\u00a0<\/em>&gt; 1, we have exponential growth; if 0 &lt; <em>b\u00a0<\/em>&lt; 1, we have exponential decay.<\/li>\r\n\t<li>We can also write this formula in terms of continuous growth as [latex]A={A}_{0}{e}^{kx}[\/latex], where [latex]{A}_{0}[\/latex] is the starting value. If [latex]{A}_{0}[\/latex] is positive, then we have exponential growth when <em>k\u00a0<\/em>&gt; 0 and exponential decay when <em>k\u00a0<\/em>&lt; 0.<\/li>\r\n\t<li>In general, we solve problems involving exponential growth or decay in two steps. First, we set up a model and use the model to find the parameters. Then we use the formula with these parameters to predict growth and decay.<\/li>\r\n\t<li>We can find the age, <em>t<\/em>, of an organic artifact by measuring the amount, <em>k<\/em>, of carbon-14 remaining in the artifact and using the formula [latex]t=\\frac{\\mathrm{ln}\\left(k\\right)}{-0.000121}[\/latex] to solve for <em>t<\/em>.<\/li>\r\n\t<li>Given a substance\u2019s doubling time or half-time, we can find a function that represents its exponential growth or decay.<\/li>\r\n\t<li>We can use Newton\u2019s Law of Cooling to find how long it will take for a cooling object to reach a desired temperature, or to find what temperature an object will be after a given time.<\/li>\r\n\t<li>We can use logistic growth functions to model real-world situations where the rate of growth changes over time, such as population growth, spread of disease, and spread of rumors.<\/li>\r\n\t<li>We can use real-world data gathered over time to observe trends. Knowledge of linear, exponential, logarithmic, and logistic graphs help us to develop models that best fit our data.<\/li>\r\n\t<li>Any exponential function with the form [latex]y=a{b}^{x}[\/latex] can be rewritten as an equivalent exponential function with the form [latex]y={A}_{0}{e}^{kx}[\/latex] where [latex]k=\\mathrm{ln}b[\/latex].<\/li>\r\n<\/ul><h2 data-type=\"glossary-title\">Glossary<\/h2>\r\n<dl id=\"fs-id1165134069388\" class=\"definition\"><dt><strong>carrying capacity<\/strong><\/dt><dd id=\"fs-id1165134069394\">in a logistic model, the limiting value of the output<\/dd><\/dl><dl id=\"fs-id1165134069398\" class=\"definition\"><dt><strong>doubling time<\/strong><\/dt><dd id=\"fs-id1165135528921\">the time it takes for a quantity to double<\/dd><\/dl><dl id=\"fs-id1165135528925\" class=\"definition\"><dt><strong>half-life<\/strong><\/dt><dd id=\"fs-id1165135528931\">the length of time it takes for a substance to exponentially decay to half of its original quantity<\/dd><\/dl><dl id=\"fs-id1165135528936\" class=\"definition\"><dt><strong>logistic growth model<\/strong><\/dt><dd id=\"fs-id1165135528941\">a function of the form [latex]f\\left(x\\right)=\\frac{c}{1+a{e}^{-bx}}[\/latex] where [latex]\\frac{c}{1+a}[\/latex] is the initial value, <em>c<\/em>\u00a0is the carrying capacity, or limiting value, and <em>b<\/em>\u00a0is a constant determined by the rate of growth<\/dd><\/dl><dl id=\"fs-id1165137844266\" class=\"definition\"><dt><strong>Newton\u2019s Law of Cooling<\/strong><\/dt><dd>the scientific formula for temperature as a function of time as an object\u2019s temperature is equalized with the ambient temperature<\/dd><\/dl><dl id=\"fs-id1165137844278\" class=\"definition\"><dt><strong>order of magnitude<\/strong><\/dt><dd id=\"fs-id1165137844283\">the power of ten, when a number is expressed in scientific notation, with one non-zero digit to the left of the decimal<\/dd><\/dl><\/section>","rendered":"<section id=\"fs-id1165135208780\" class=\"key-equations\" data-depth=\"1\">\n<h2 data-type=\"title\">Key Equations<\/h2>\n<table id=\"fs-id2347772\" summary=\"..\">\n<tbody>\n<tr>\n<td>Half-life formula<\/td>\n<td>If [latex]\\text{ }A={A}_{0}{e}^{kt}[\/latex], <em>k\u00a0<\/em>&lt; 0, the half-life is [latex]t=-\\frac{\\mathrm{ln}\\left(2\\right)}{k}[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>Carbon-14 dating<\/td>\n<td>\n<p>[latex]t=\\frac{\\mathrm{ln}\\left(\\frac{A}{{A}_{0}}\\right)}{-0.000121}[\/latex].[latex]{A}_{0}[\/latex] <em>A<\/em>\u00a0is the amount of carbon-14 when the plant or animal died<\/p>\n<p><em>t<\/em>\u00a0is the amount of carbon-14 remaining today<\/p>\n<p>is the age of the fossil in years<\/td>\n<\/tr>\n<tr>\n<td>Doubling time formula<\/td>\n<td>If [latex]A={A}_{0}{e}^{kt}[\/latex], <em>k\u00a0<\/em>&gt; 0, the doubling time is [latex]t=\\frac{\\mathrm{ln}2}{k}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Newton\u2019s Law of Cooling<\/td>\n<td>[latex]T\\left(t\\right)=A{e}^{kt}+{T}_{s}[\/latex], where [latex]{T}_{s}[\/latex] is the ambient temperature, [latex]A=T\\left(0\\right)-{T}_{s}[\/latex], and <em>k<\/em> is the continuous rate of cooling.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/section>\n<section id=\"fs-id1165137894245\" class=\"key-concepts\" data-depth=\"1\">\n<h2 data-type=\"title\">Key Concepts<\/h2>\n<ul id=\"fs-id1165137894248\">\n<li>The basic exponential function is [latex]f\\left(x\\right)=a{b}^{x}[\/latex]. If <em>b\u00a0<\/em>&gt; 1, we have exponential growth; if 0 &lt; <em>b\u00a0<\/em>&lt; 1, we have exponential decay.<\/li>\n<li>We can also write this formula in terms of continuous growth as [latex]A={A}_{0}{e}^{kx}[\/latex], where [latex]{A}_{0}[\/latex] is the starting value. If [latex]{A}_{0}[\/latex] is positive, then we have exponential growth when <em>k\u00a0<\/em>&gt; 0 and exponential decay when <em>k\u00a0<\/em>&lt; 0.<\/li>\n<li>In general, we solve problems involving exponential growth or decay in two steps. First, we set up a model and use the model to find the parameters. Then we use the formula with these parameters to predict growth and decay.<\/li>\n<li>We can find the age, <em>t<\/em>, of an organic artifact by measuring the amount, <em>k<\/em>, of carbon-14 remaining in the artifact and using the formula [latex]t=\\frac{\\mathrm{ln}\\left(k\\right)}{-0.000121}[\/latex] to solve for <em>t<\/em>.<\/li>\n<li>Given a substance\u2019s doubling time or half-time, we can find a function that represents its exponential growth or decay.<\/li>\n<li>We can use Newton\u2019s Law of Cooling to find how long it will take for a cooling object to reach a desired temperature, or to find what temperature an object will be after a given time.<\/li>\n<li>We can use logistic growth functions to model real-world situations where the rate of growth changes over time, such as population growth, spread of disease, and spread of rumors.<\/li>\n<li>We can use real-world data gathered over time to observe trends. Knowledge of linear, exponential, logarithmic, and logistic graphs help us to develop models that best fit our data.<\/li>\n<li>Any exponential function with the form [latex]y=a{b}^{x}[\/latex] can be rewritten as an equivalent exponential function with the form [latex]y={A}_{0}{e}^{kx}[\/latex] where [latex]k=\\mathrm{ln}b[\/latex].<\/li>\n<\/ul>\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl id=\"fs-id1165134069388\" class=\"definition\">\n<dt><strong>carrying capacity<\/strong><\/dt>\n<dd id=\"fs-id1165134069394\">in a logistic model, the limiting value of the output<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134069398\" class=\"definition\">\n<dt><strong>doubling time<\/strong><\/dt>\n<dd id=\"fs-id1165135528921\">the time it takes for a quantity to double<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135528925\" class=\"definition\">\n<dt><strong>half-life<\/strong><\/dt>\n<dd id=\"fs-id1165135528931\">the length of time it takes for a substance to exponentially decay to half of its original quantity<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135528936\" class=\"definition\">\n<dt><strong>logistic growth model<\/strong><\/dt>\n<dd id=\"fs-id1165135528941\">a function of the form [latex]f\\left(x\\right)=\\frac{c}{1+a{e}^{-bx}}[\/latex] where [latex]\\frac{c}{1+a}[\/latex] is the initial value, <em>c<\/em>\u00a0is the carrying capacity, or limiting value, and <em>b<\/em>\u00a0is a constant determined by the rate of growth<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137844266\" class=\"definition\">\n<dt><strong>Newton\u2019s Law of Cooling<\/strong><\/dt>\n<dd>the scientific formula for temperature as a function of time as an object\u2019s temperature is equalized with the ambient temperature<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137844278\" class=\"definition\">\n<dt><strong>order of magnitude<\/strong><\/dt>\n<dd id=\"fs-id1165137844283\">the power of ten, when a number is expressed in scientific notation, with one non-zero digit to the left of the decimal<\/dd>\n<\/dl>\n<\/section>\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-1685\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: Jay Abramson, et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.<\/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":276,"menu_order":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"Jay Abramson, et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1685","chapter","type-chapter","status-publish","hentry"],"part":1668,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1685","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":3,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1685\/revisions"}],"predecessor-version":[{"id":3091,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1685\/revisions\/3091"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/1668"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1685\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/media?parent=1685"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1685"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/contributor?post=1685"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/license?post=1685"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}