{"id":2146,"date":"2024-07-10T01:24:13","date_gmt":"2024-07-10T01:24:13","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=2146"},"modified":"2025-01-22T05:04:32","modified_gmt":"2025-01-22T05:04:32","slug":"applications-of-exponential-functions-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/applications-of-exponential-functions-fresh-take\/","title":{"raw":"Applications of Exponential Functions: Fresh Take","rendered":"Applications of Exponential Functions: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Calculate the values of exponential functions, especially those using the base \ud835\udc52, and understand their equations<\/li>\r\n \t<li>Use compound interest formulas to work out how investments or loans grow over time in real-life financial situations<\/li>\r\n \t<li>Find an exponential function that models continuous growth or decay<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Compound Interest<\/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\">Compound Interest: Interest earned on both the principal and previously accumulated interest.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Annual Percentage Rate (APR): The nominal yearly interest rate.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Compounding Frequency: How often interest is calculated and added to the principal.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Annual Percentage Yield (APY): The effective annual rate of return, taking compounding into account.<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Key Formulas<\/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\">Compound Interest Formula: [latex]A(t) = P(1 + \\frac{r}{n})^{nt}[\/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]A(t)[\/latex] = Final amount<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]P[\/latex] = Principal (initial investment)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]r[\/latex] = Annual interest rate (as a decimal)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]n[\/latex] = Number of times interest is compounded per year<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]t[\/latex] = Number of years<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Annual Percentage Yield (APY) Formula: [latex]APY = (1 + \\frac{r}{n})^n - 1[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">An initial investment of [latex]$100,000[\/latex] at [latex]12 \\%[\/latex] interest is compounded weekly (use [latex]52[\/latex] weeks in a year). What will the investment be worth in [latex]30[\/latex] years?[reveal-answer q=\"939452\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"939452\"]about [latex]$3,644,675.88[\/latex][\/hidden-answer]<\/section>\r\n<h2>Evaluating Exponential Functions with Base [latex]e[\/latex]<\/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\">The Number [latex]e[\/latex]: An irrational number approximately equal to [latex]2.718282[\/latex].<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Definition of [latex]e[\/latex]: [latex]e = \\lim_{n \\to \\infty} (1 + \\frac{1}{n})^n[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Natural Exponential Function: [latex]f(x) = e^x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Applications: Widely used in modeling natural phenomena and financial calculations.<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Key Properties of [latex]f(x) = e^x[\/latex]<\/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\">Domain: All real numbers [latex](-\\infty, \\infty)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Range: All positive real numbers [latex](0, \\infty)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]y[\/latex]-intercept: ([latex]0, 1)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Horizontal asymptote: [latex]y = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Always increasing<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]e^0 = 1[\/latex]<\/li>\r\n<\/ol>\r\n<strong>\u00a0<\/strong>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Use a calculator to find [latex]{e}^{-0.5}[\/latex]. Round to five decimal places.[reveal-answer q=\"168744\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"168744\"][latex]{e}^{-0.5}\\approx 0.60653[\/latex][\/hidden-answer]<\/section><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-cfbdbfdg-XgVnygbLnJw\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/XgVnygbLnJw?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-cfbdbfdg-XgVnygbLnJw\" 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=12850335&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-cfbdbfdg-XgVnygbLnJw&vembed=0&video_id=XgVnygbLnJw&video_target=tpm-plugin-cfbdbfdg-XgVnygbLnJw'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Compounded+Interest_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cCompounded Interest\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><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-fagfgecf-r26vL3XLnCQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/r26vL3XLnCQ?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-fagfgecf-r26vL3XLnCQ\" 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=12850337&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-fagfgecf-r26vL3XLnCQ&vembed=0&video_id=r26vL3XLnCQ&video_target=tpm-plugin-fagfgecf-r26vL3XLnCQ'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+2+-+Continuous+Interest+with+Logarithms_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 2: Continuous Interest with Logarithms\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Investigating Continuous Growth<\/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\">Continuous Growth\/Decay Formula: [latex]A(t) = Pe^{rt}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Components:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]P[\/latex]: Initial value or principal<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]r[\/latex]: Growth or decay rate per unit time<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]t[\/latex]: Time period<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]e[\/latex]: Mathematical constant ([latex]\u2248 2.71828[\/latex])<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Types:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Continuous Growth: [latex]r &gt; 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Continuous Decay: [latex]r &lt; 0[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Key Properties<\/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\">Exponential nature: Growth\/decay occurs continuously<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Compound interest limit: As compounding frequency approaches infinity<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Time independence: Rate of change proportional to current value<\/li>\r\n<\/ol>\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 initial value ([latex]P[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Determine growth\/decay rate ([latex]r[\/latex])\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Ensure [latex]r[\/latex] is expressed as a decimal<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use negative [latex]r[\/latex] for decay<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Identify time period ([latex]t[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Substitute values into [latex]A(t) = Pe^{rt}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculate final value using a calculator<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">A person invests [latex]$100,000[\/latex] at a nominal [latex]12 \\%[\/latex] interest per year compounded continuously. What will be the value of the investment in [latex]30[\/latex] years?[reveal-answer q=\"59872\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"59872\"][latex]$3,659,823.44[\/latex][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Radon-222 decays at a continuous rate of [latex]17.3 \\%[\/latex] per day. How much will [latex]100[\/latex] mg of Radon-[latex]222[\/latex] remain after one year?[reveal-answer q=\"58534\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"58534\"][latex]3.77115984\\small{E }-26[\/latex] (This is calculator notation for the number written as [latex]3.77\\times {10}^{-26}[\/latex] in scientific notation. While the output of an exponential function is never zero, this number is so close to zero that for all practical purposes we can accept zero as the answer.)[\/hidden-answer]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Calculate the values of exponential functions, especially those using the base \ud835\udc52, and understand their equations<\/li>\n<li>Use compound interest formulas to work out how investments or loans grow over time in real-life financial situations<\/li>\n<li>Find an exponential function that models continuous growth or decay<\/li>\n<\/ul>\n<\/section>\n<h2>Compound Interest<\/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\">Compound Interest: Interest earned on both the principal and previously accumulated interest.<\/li>\n<li class=\"whitespace-normal break-words\">Annual Percentage Rate (APR): The nominal yearly interest rate.<\/li>\n<li class=\"whitespace-normal break-words\">Compounding Frequency: How often interest is calculated and added to the principal.<\/li>\n<li class=\"whitespace-normal break-words\">Annual Percentage Yield (APY): The effective annual rate of return, taking compounding into account.<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Key Formulas<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Compound Interest Formula: [latex]A(t) = P(1 + \\frac{r}{n})^{nt}[\/latex] Where:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]A(t)[\/latex] = Final amount<\/li>\n<li class=\"whitespace-normal break-words\">[latex]P[\/latex] = Principal (initial investment)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]r[\/latex] = Annual interest rate (as a decimal)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]n[\/latex] = Number of times interest is compounded per year<\/li>\n<li class=\"whitespace-normal break-words\">[latex]t[\/latex] = Number of years<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Annual Percentage Yield (APY) Formula: [latex]APY = (1 + \\frac{r}{n})^n - 1[\/latex]<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">An initial investment of [latex]$100,000[\/latex] at [latex]12 \\%[\/latex] interest is compounded weekly (use [latex]52[\/latex] weeks in a year). What will the investment be worth in [latex]30[\/latex] years?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q939452\">Show Solution<\/button><\/p>\n<div id=\"q939452\" class=\"hidden-answer\" style=\"display: none\">about [latex]$3,644,675.88[\/latex]<\/div>\n<\/div>\n<\/section>\n<h2>Evaluating Exponential Functions with Base [latex]e[\/latex]<\/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\">The Number [latex]e[\/latex]: An irrational number approximately equal to [latex]2.718282[\/latex].<\/li>\n<li class=\"whitespace-normal break-words\">Definition of [latex]e[\/latex]: [latex]e = \\lim_{n \\to \\infty} (1 + \\frac{1}{n})^n[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Natural Exponential Function: [latex]f(x) = e^x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Applications: Widely used in modeling natural phenomena and financial calculations.<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Key Properties of [latex]f(x) = e^x[\/latex]<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Domain: All real numbers [latex](-\\infty, \\infty)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Range: All positive real numbers [latex](0, \\infty)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]y[\/latex]-intercept: ([latex]0, 1)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Horizontal asymptote: [latex]y = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Always increasing<\/li>\n<li class=\"whitespace-normal break-words\">[latex]e^0 = 1[\/latex]<\/li>\n<\/ol>\n<p><strong>\u00a0<\/strong><\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Use a calculator to find [latex]{e}^{-0.5}[\/latex]. Round to five decimal places.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q168744\">Show Solution<\/button><\/p>\n<div id=\"q168744\" class=\"hidden-answer\" style=\"display: none\">[latex]{e}^{-0.5}\\approx 0.60653[\/latex]<\/div>\n<\/div>\n<\/section>\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-cfbdbfdg-XgVnygbLnJw\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/XgVnygbLnJw?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-cfbdbfdg-XgVnygbLnJw\" 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=12850335&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-cfbdbfdg-XgVnygbLnJw&#38;vembed=0&#38;video_id=XgVnygbLnJw&#38;video_target=tpm-plugin-cfbdbfdg-XgVnygbLnJw\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Compounded+Interest_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cCompounded Interest\u201d here (opens in new window).<\/a><\/p>\n<\/section>\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-fagfgecf-r26vL3XLnCQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/r26vL3XLnCQ?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-fagfgecf-r26vL3XLnCQ\" 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=12850337&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-fagfgecf-r26vL3XLnCQ&#38;vembed=0&#38;video_id=r26vL3XLnCQ&#38;video_target=tpm-plugin-fagfgecf-r26vL3XLnCQ\"><\/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+2+-+Continuous+Interest+with+Logarithms_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 2: Continuous Interest with Logarithms\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Investigating Continuous Growth<\/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\">Continuous Growth\/Decay Formula: [latex]A(t) = Pe^{rt}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Components:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]P[\/latex]: Initial value or principal<\/li>\n<li class=\"whitespace-normal break-words\">[latex]r[\/latex]: Growth or decay rate per unit time<\/li>\n<li class=\"whitespace-normal break-words\">[latex]t[\/latex]: Time period<\/li>\n<li class=\"whitespace-normal break-words\">[latex]e[\/latex]: Mathematical constant ([latex]\u2248 2.71828[\/latex])<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Types:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Continuous Growth: [latex]r > 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Continuous Decay: [latex]r < 0[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Key Properties<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Exponential nature: Growth\/decay occurs continuously<\/li>\n<li class=\"whitespace-normal break-words\">Compound interest limit: As compounding frequency approaches infinity<\/li>\n<li class=\"whitespace-normal break-words\">Time independence: Rate of change proportional to current value<\/li>\n<\/ol>\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 initial value ([latex]P[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">Determine growth\/decay rate ([latex]r[\/latex])\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Ensure [latex]r[\/latex] is expressed as a decimal<\/li>\n<li class=\"whitespace-normal break-words\">Use negative [latex]r[\/latex] for decay<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Identify time period ([latex]t[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">Substitute values into [latex]A(t) = Pe^{rt}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Calculate final value using a calculator<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">A person invests [latex]$100,000[\/latex] at a nominal [latex]12 \\%[\/latex] interest per year compounded continuously. What will be the value of the investment in [latex]30[\/latex] years?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q59872\">Show Solution<\/button><\/p>\n<div id=\"q59872\" class=\"hidden-answer\" style=\"display: none\">[latex]$3,659,823.44[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Radon-222 decays at a continuous rate of [latex]17.3 \\%[\/latex] per day. How much will [latex]100[\/latex] mg of Radon-[latex]222[\/latex] remain after one year?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q58534\">Show Solution<\/button><\/p>\n<div id=\"q58534\" class=\"hidden-answer\" style=\"display: none\">[latex]3.77115984\\small{E }-26[\/latex] (This is calculator notation for the number written as [latex]3.77\\times {10}^{-26}[\/latex] in scientific notation. While the output of an exponential function is never zero, this number is so close to zero that for all practical purposes we can accept zero as the answer.)<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":12,"menu_order":18,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Compounded Interest\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/XgVnygbLnJw\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex 2:  Continuous Interest with Logarithms\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/r26vL3XLnCQ\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":255,"module-header":"fresh_take","content_attributions":[{"type":"copyrighted_video","description":"Compounded Interest","author":"","organization":"Mathispower4u","url":"https:\/\/youtu.be\/XgVnygbLnJw","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"Ex 2:  Continuous Interest with Logarithms","author":"","organization":"Mathispower4u","url":"https:\/\/youtu.be\/r26vL3XLnCQ","project":"","license":"arr","license_terms":"Standard YouTube License"}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><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=12850335&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-cfbdbfdg-XgVnygbLnJw&vembed=0&video_id=XgVnygbLnJw&video_target=tpm-plugin-cfbdbfdg-XgVnygbLnJw'><\/script>\n<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><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=12850337&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-fagfgecf-r26vL3XLnCQ&vembed=0&video_id=r26vL3XLnCQ&video_target=tpm-plugin-fagfgecf-r26vL3XLnCQ'><\/script>\n","media_targets":["tpm-plugin-cfbdbfdg-XgVnygbLnJw","tpm-plugin-fagfgecf-r26vL3XLnCQ"]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2146"}],"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":5,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2146\/revisions"}],"predecessor-version":[{"id":7282,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2146\/revisions\/7282"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/255"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2146\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=2146"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2146"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=2146"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=2146"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}