{"id":3159,"date":"2025-08-15T22:22:34","date_gmt":"2025-08-15T22:22:34","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=3159"},"modified":"2026-03-17T17:32:59","modified_gmt":"2026-03-17T17:32:59","slug":"exponential-functions-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/exponential-functions-apply-it\/","title":{"raw":"Exponential Functions: Apply It 1","rendered":"Exponential Functions: Apply It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Evaluate exponential functions.<\/li>\r\n \t<li>Find the equation of an exponential function.<\/li>\r\n \t<li>Use compound interest formulas.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Applications of Exponential Functions<\/h2>\r\n<strong>Exponential functions<\/strong> are incredibly powerful tools in mathematics, and they have a wide range of applications in the real world. Whether you're looking at population growth, radioactive decay, or even finance, exponential functions help us model situations where change happens at a constant multiplicative rate.\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n\r\n[caption id=\"\" align=\"alignright\" width=\"250\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02225437\/CNX_Precalc_Figure_04_00_001.jpg\" alt=\"Escherichia coli (e Coli) bacteria\" width=\"250\" height=\"180\" \/> An electron micrograph of E.Coli bacteria. (credit: \u201cMattosaurus,\u201d Wikimedia Commons)[\/caption]\r\n\r\nBacteria commonly reproduce through a process called binary fission during which one bacterial cell splits into two. When conditions are right, bacteria can reproduce very quickly. Unlike humans and other complex organisms, the time required to form a new generation of bacteria is often a matter of minutes or hours as opposed to days or years.[footnote]Todar, PhD, Kenneth. Todar's Online Textbook of Bacteriology. http:\/\/textbookofbacteriology.net\/growth_3.html.[\/footnote]\r\n\r\nFor simplicity\u2019s sake, suppose we begin with a culture of one bacterial cell that can divide every hour. The table below\u00a0shows the number of bacterial cells at the end of each subsequent hour. We see that the single bacterial cell leads to over one thousand bacterial cells in just ten hours! If we were to extrapolate the table to twenty-four hours, we would have over [latex]16[\/latex] million!\r\n<table id=\"Table_04_00_01\" summary=\"\">\r\n<tbody>\r\n<tr>\r\n<td><strong>Hour<\/strong><\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]5[\/latex]<\/td>\r\n<td>[latex]6[\/latex]<\/td>\r\n<td>[latex]7[\/latex]<\/td>\r\n<td>[latex]8[\/latex]<\/td>\r\n<td>[latex]9[\/latex]<\/td>\r\n<td>[latex]10[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Bacteria<\/strong><\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]8[\/latex]<\/td>\r\n<td>[latex]16[\/latex]<\/td>\r\n<td>[latex]32[\/latex]<\/td>\r\n<td>[latex]64[\/latex]<\/td>\r\n<td>[latex]128[\/latex]<\/td>\r\n<td>[latex]256[\/latex]<\/td>\r\n<td>[latex]512[\/latex]<\/td>\r\n<td>[latex]1024[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/section><section class=\"textbox recall\" aria-label=\"Recall\">A function that models <strong>exponential growth<\/strong> grows by a rate proportional to the amount present. For any real number [latex]x[\/latex]\u00a0and any positive real numbers [latex]a[\/latex]\u00a0and [latex]b[\/latex]\u00a0such that [latex]b\\ne 1[\/latex], an exponential growth function has the form\r\n<p style=\"text-align: center;\">[latex]\\text{ }f\\left(x\\right)=a{b}^{x}[\/latex]<\/p>\r\nwhere\r\n<ul>\r\n \t<li>[latex]a[\/latex]\u00a0is the initial or starting value of the function.<\/li>\r\n \t<li>[latex]b[\/latex]\u00a0is the growth factor or growth multiplier per unit [latex]x[\/latex].<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">The population of India was about [latex]1.25[\/latex] billion in the year 2013 with an annual growth rate of about [latex]1.2 \\%[\/latex]. This situation is represented by the growth function [latex]P\\left(t\\right)=1.25{\\left(1.012\\right)}^{t}[\/latex] where [latex]t[\/latex] is the number of years since 2013. To the nearest thousandth, what will the population of India be in 2031?[reveal-answer q=\"924755\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"924755\"]To estimate the population in 2031, we evaluate the models for [latex]t\u00a0= 18[\/latex], because 2031 is [latex]18[\/latex] years after 2013. Rounding to the nearest thousandth,\r\n<p style=\"text-align: center;\">[latex]P\\left(18\\right)=1.25{\\left(1.012\\right)}^{18}\\approx 1.549[\/latex]<\/p>\r\nThere will be about [latex]1.549[\/latex] billion people in India in the year 2031.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]321416[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Evaluate exponential functions.<\/li>\n<li>Find the equation of an exponential function.<\/li>\n<li>Use compound interest formulas.<\/li>\n<\/ul>\n<\/section>\n<h2>Applications of Exponential Functions<\/h2>\n<p><strong>Exponential functions<\/strong> are incredibly powerful tools in mathematics, and they have a wide range of applications in the real world. Whether you&#8217;re looking at population growth, radioactive decay, or even finance, exponential functions help us model situations where change happens at a constant multiplicative rate.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<figure style=\"width: 250px\" class=\"wp-caption alignright\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02225437\/CNX_Precalc_Figure_04_00_001.jpg\" alt=\"Escherichia coli (e Coli) bacteria\" width=\"250\" height=\"180\" \/><figcaption class=\"wp-caption-text\">An electron micrograph of E.Coli bacteria. (credit: \u201cMattosaurus,\u201d Wikimedia Commons)<\/figcaption><\/figure>\n<p>Bacteria commonly reproduce through a process called binary fission during which one bacterial cell splits into two. When conditions are right, bacteria can reproduce very quickly. Unlike humans and other complex organisms, the time required to form a new generation of bacteria is often a matter of minutes or hours as opposed to days or years.<a class=\"footnote\" title=\"Todar, PhD, Kenneth. Todar's Online Textbook of Bacteriology. http:\/\/textbookofbacteriology.net\/growth_3.html.\" id=\"return-footnote-3159-1\" href=\"#footnote-3159-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a><\/p>\n<p>For simplicity\u2019s sake, suppose we begin with a culture of one bacterial cell that can divide every hour. The table below\u00a0shows the number of bacterial cells at the end of each subsequent hour. We see that the single bacterial cell leads to over one thousand bacterial cells in just ten hours! If we were to extrapolate the table to twenty-four hours, we would have over [latex]16[\/latex] million!<\/p>\n<table id=\"Table_04_00_01\" summary=\"\">\n<tbody>\n<tr>\n<td><strong>Hour<\/strong><\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]5[\/latex]<\/td>\n<td>[latex]6[\/latex]<\/td>\n<td>[latex]7[\/latex]<\/td>\n<td>[latex]8[\/latex]<\/td>\n<td>[latex]9[\/latex]<\/td>\n<td>[latex]10[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Bacteria<\/strong><\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]8[\/latex]<\/td>\n<td>[latex]16[\/latex]<\/td>\n<td>[latex]32[\/latex]<\/td>\n<td>[latex]64[\/latex]<\/td>\n<td>[latex]128[\/latex]<\/td>\n<td>[latex]256[\/latex]<\/td>\n<td>[latex]512[\/latex]<\/td>\n<td>[latex]1024[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/section>\n<section class=\"textbox recall\" aria-label=\"Recall\">A function that models <strong>exponential growth<\/strong> grows by a rate proportional to the amount present. For any real number [latex]x[\/latex]\u00a0and any positive real numbers [latex]a[\/latex]\u00a0and [latex]b[\/latex]\u00a0such that [latex]b\\ne 1[\/latex], an exponential growth function has the form<\/p>\n<p style=\"text-align: center;\">[latex]\\text{ }f\\left(x\\right)=a{b}^{x}[\/latex]<\/p>\n<p>where<\/p>\n<ul>\n<li>[latex]a[\/latex]\u00a0is the initial or starting value of the function.<\/li>\n<li>[latex]b[\/latex]\u00a0is the growth factor or growth multiplier per unit [latex]x[\/latex].<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">The population of India was about [latex]1.25[\/latex] billion in the year 2013 with an annual growth rate of about [latex]1.2 \\%[\/latex]. This situation is represented by the growth function [latex]P\\left(t\\right)=1.25{\\left(1.012\\right)}^{t}[\/latex] where [latex]t[\/latex] is the number of years since 2013. To the nearest thousandth, what will the population of India be in 2031?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q924755\">Show Solution<\/button><\/p>\n<div id=\"q924755\" class=\"hidden-answer\" style=\"display: none\">To estimate the population in 2031, we evaluate the models for [latex]t\u00a0= 18[\/latex], because 2031 is [latex]18[\/latex] years after 2013. Rounding to the nearest thousandth,<\/p>\n<p style=\"text-align: center;\">[latex]P\\left(18\\right)=1.25{\\left(1.012\\right)}^{18}\\approx 1.549[\/latex]<\/p>\n<p>There will be about [latex]1.549[\/latex] billion people in India in the year 2031.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm321416\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=321416&theme=lumen&iframe_resize_id=ohm321416&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-3159-1\">Todar, PhD, Kenneth. Todar's Online Textbook of Bacteriology. http:\/\/textbookofbacteriology.net\/growth_3.html. <a href=\"#return-footnote-3159-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":67,"menu_order":14,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":105,"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\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3159"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":4,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3159\/revisions"}],"predecessor-version":[{"id":5879,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3159\/revisions\/5879"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/105"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3159\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=3159"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=3159"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=3159"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=3159"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}