{"id":5602,"date":"2021-02-05T22:21:34","date_gmt":"2021-02-05T22:21:34","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/?post_type=chapter&#038;p=5602"},"modified":"2025-11-30T03:20:47","modified_gmt":"2025-11-30T03:20:47","slug":"3-3-exponential-growth","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/chapter\/3-3-exponential-growth\/","title":{"raw":"3.3 Exponential Growth","rendered":"3.3 Exponential Growth"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Apply and construct exponential models using the form y = a(b)<sup>x <\/sup><\/li>\r\n<\/ul>\r\n<\/div>\r\nJust as in linear growth, we can model quantities that grow in a non-linear pattern. Consider exponential growth.\r\n<p style=\"text-align: center;\"><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/02\/05224552\/pennies.jpg\"><img class=\"alignnone wp-image-5610\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/02\/05224552\/pennies-300x198.jpg\" alt=\"\" width=\"486\" height=\"321\" \/><\/a><\/p>\r\n\r\n<h3>Exponential Models<\/h3>\r\nConsider the older brother who offered to pay his younger brother 1 penny per day to help him mow lawns and rake leaves. The younger brother said, \"No thanks! But, I will work for you for 30 days in a row. You will pay me 1 penny on the first day and then each day after that you will double that amount you paid me the previous day.\"\r\n\r\nLet's model that situation with a table and a graph. Let x represent the number of days, with zero representing Day 1. Let y = the amount of money the younger brother earns in this model.\r\n<table style=\"width: 381px;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 67.75px; text-align: left;\"><strong>Day<\/strong><\/td>\r\n<td style=\"width: 227.15px;\"><strong>x = number of days (Day 1 is 0)<\/strong><\/td>\r\n<td style=\"width: 165.3px;\"><strong>y = amount of money<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 67.75px;\">1<\/td>\r\n<td style=\"width: 227.15px;\">0<\/td>\r\n<td style=\"width: 165.3px;\">$0.01<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 67.75px;\">2<\/td>\r\n<td style=\"width: 227.15px;\">1<\/td>\r\n<td style=\"width: 165.3px;\">$0.02<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 67.75px;\">3<\/td>\r\n<td style=\"width: 227.15px;\">2<\/td>\r\n<td style=\"width: 165.3px;\">$0.04<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 67.75px;\">4<\/td>\r\n<td style=\"width: 227.15px;\">3<\/td>\r\n<td style=\"width: 165.3px;\">$0.08<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 67.75px;\">5<\/td>\r\n<td style=\"width: 227.15px;\">4<\/td>\r\n<td style=\"width: 165.3px;\">$0.16<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 67.75px;\">10<\/td>\r\n<td style=\"width: 227.15px;\">9<\/td>\r\n<td style=\"width: 165.3px;\">$10.24<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 67.75px;\">25<\/td>\r\n<td style=\"width: 227.15px;\">24<\/td>\r\n<td style=\"width: 165.3px;\">$167,772.16<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 67.75px;\">26<\/td>\r\n<td style=\"width: 227.15px;\">25<\/td>\r\n<td style=\"width: 165.3px;\">$333,544<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 67.75px;\">27<\/td>\r\n<td style=\"width: 227.15px;\">26<\/td>\r\n<td style=\"width: 165.3px;\">$671,088.64<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 67.75px;\">28<\/td>\r\n<td style=\"width: 227.15px;\">27<\/td>\r\n<td style=\"width: 165.3px;\">$1,342,177.28<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 67.75px;\">29<\/td>\r\n<td style=\"width: 227.15px;\">28<\/td>\r\n<td style=\"width: 165.3px;\">$2,684,354.56<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 67.75px;\">30<\/td>\r\n<td style=\"width: 227.15px;\">29<\/td>\r\n<td style=\"width: 165.3px;\">$5,368,709.12<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/02\/06213343\/Expo-1.jpg\"><img class=\"wp-image-5614 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/02\/06213343\/Expo-1.jpg\" alt=\"\" width=\"679\" height=\"521\" \/><\/a>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\nNotice the table of x- and y-values, along with the graph. Younger brother would only be making $10.24 on Day 10. But, then something happens around Day 25. The pay drastically increases. By Day 30, he would be making over $5 million.\r\n\r\nGraphs shaped like this are modeling <strong>exponential growth<\/strong>. Let's examine the table to determine the equation that models the younger brother's pay.\r\n<table style=\"width: 979px; height: 196px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 28px;\">\r\n<td style=\"width: 51.8667px; height: 28px;\"><strong>Day<\/strong><\/td>\r\n<td style=\"width: 120.75px; height: 28px;\"><strong>x = number of days (Day 1 is 0)<\/strong><\/td>\r\n<td style=\"width: 510.533px; height: 28px;\"><strong>Pattern<\/strong><\/td>\r\n<td style=\"width: 245.133px; height: 28px;\"><strong>y = amount of money<\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 51.8667px; height: 14px;\">1<\/td>\r\n<td style=\"width: 120.75px; height: 14px;\">0<\/td>\r\n<td style=\"width: 510.533px; height: 14px;\">Pay on Day 1 = $0.01<\/td>\r\n<td style=\"width: 245.133px; height: 14px;\">$0.01<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 51.8667px; height: 14px;\">2<\/td>\r\n<td style=\"width: 120.75px; height: 14px;\">1<\/td>\r\n<td style=\"width: 510.533px; height: 14px;\">Pay on Day 2 = $0.01 x 2 \u00a0 \u00a0 (When something doubles, that means multiply by 2.)<\/td>\r\n<td style=\"width: 245.133px; height: 14px;\">$0.02<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 51.8667px; height: 14px;\">3<\/td>\r\n<td style=\"width: 120.75px; height: 14px;\">2<\/td>\r\n<td style=\"width: 510.533px; height: 14px;\">Pay on Day 3 = $0.01 x 2 x 2<\/td>\r\n<td style=\"width: 245.133px; height: 14px;\">$0.04<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 51.8667px; height: 14px;\">4<\/td>\r\n<td style=\"width: 120.75px; height: 14px;\">3<\/td>\r\n<td style=\"width: 510.533px; height: 14px;\">Pay on Day 4 = $0.01 x 2<sup>3<\/sup> \u00a0\u00a0 (Switch notation to using exponents.)<\/td>\r\n<td style=\"width: 245.133px; height: 14px;\">$0.08<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 51.8667px; height: 14px;\">5<\/td>\r\n<td style=\"width: 120.75px; height: 14px;\">4<\/td>\r\n<td style=\"width: 510.533px; height: 14px;\">Pay on Day 5 = $0.01 x 2<sup>4<\/sup><\/td>\r\n<td style=\"width: 245.133px; height: 14px;\">$0.16<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 51.8667px; height: 14px;\">10<\/td>\r\n<td style=\"width: 120.75px; height: 14px;\">9<\/td>\r\n<td style=\"width: 510.533px; height: 14px;\">Pay on Day 10 = $0.01 x 2<sup>9<\/sup><\/td>\r\n<td style=\"width: 245.133px; height: 14px;\">$10.24<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 51.8667px; height: 14px;\">25<\/td>\r\n<td style=\"width: 120.75px; height: 14px;\">24<\/td>\r\n<td style=\"width: 510.533px; height: 14px;\">Pay on Day 25 = $0.01 x 2<sup>24<\/sup><\/td>\r\n<td style=\"width: 245.133px; height: 14px;\">$167,772.16<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 51.8667px; height: 14px;\">26<\/td>\r\n<td style=\"width: 120.75px; height: 14px;\">25<\/td>\r\n<td style=\"width: 510.533px; height: 14px;\">Pay on Day 26 = $0.01 x 2<sup>25<\/sup><\/td>\r\n<td style=\"width: 245.133px; height: 14px;\">$333,544<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 51.8667px; height: 14px;\">27<\/td>\r\n<td style=\"width: 120.75px; height: 14px;\">26<\/td>\r\n<td style=\"width: 510.533px; height: 14px;\">Pay on Day 27 = $0.01 x 2<sup>26<\/sup><\/td>\r\n<td style=\"width: 245.133px; height: 14px;\">$671,088.64<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 51.8667px; height: 14px;\">28<\/td>\r\n<td style=\"width: 120.75px; height: 14px;\">27<\/td>\r\n<td style=\"width: 510.533px; height: 14px;\">Pay on Day 28 = $0.01 x 2<sup>27<\/sup><\/td>\r\n<td style=\"width: 245.133px; height: 14px;\">$1,342,177.28<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 51.8667px; height: 14px;\">29<\/td>\r\n<td style=\"width: 120.75px; height: 14px;\">28<\/td>\r\n<td style=\"width: 510.533px; height: 14px;\">Pay on Day 29 = $0.01 x 2<sup>28<\/sup><\/td>\r\n<td style=\"width: 245.133px; height: 14px;\">$2,684,354.56<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 51.8667px; height: 14px;\">30<\/td>\r\n<td style=\"width: 120.75px; height: 14px;\">29<\/td>\r\n<td style=\"width: 510.533px; height: 14px;\">Pay on Day 30 = $0.01 x 2<sup>29<\/sup><\/td>\r\n<td style=\"width: 245.133px; height: 14px;\">$5,368,709.12<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nLetting x be the number of days after the first day, then Day 1 is 0 and Day 2 is 1, and so on. Then letting y be the amount of pay younger brother is going to receive, the last day of the month younger brother receives $0.01 x 2<sup>29<\/sup>. Now, letting x represent any day, the equation becomes: <strong>y = $0.01(2)<sup>x <\/sup><\/strong>.\r\n\r\nSuppose that the younger brother said the pay would be <strong>triple<\/strong> the amount paid on the previous day. The model would be:\r\n<p style=\"text-align: center;\"><strong>y = $0.01(3)<sup>x<\/sup><\/strong><\/p>\r\nIf the younger brother had used the word <strong>quadruple<\/strong>, the model would be:\r\n<p style=\"text-align: center;\"><strong>y = $0.01(4)<sup>x<\/sup><\/strong><\/p>\r\n\r\n<h3>Investing Money<\/h3>\r\nConsider the scenario of investing money.\r\n\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/02\/07000134\/money.jpg\"><img class=\"wp-image-5622 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/02\/07000134\/money-300x203.jpg\" alt=\"\" width=\"476\" height=\"322\" \/><\/a>\r\n\r\nSuppose $5,000 is invested into an investment that earns 20% simple interest per year. To compute the interest earned in the first year, use the simple interest formula: <strong>Interest = Principle x Rate x Time (in years), <\/strong>or<strong> I = PRT.<\/strong>\r\n\r\nAfter one year, the interest earned would be:\r\n<p style=\"padding-left: 90px;\">I = PRT<\/p>\r\n<p style=\"padding-left: 90px;\">\u00a0 = ($5,000) x (0.20) x (1 year)<\/p>\r\n<p style=\"padding-left: 90px;\">\u00a0 = $1,000<\/p>\r\nThen, adding the interest earned in the first year ($1,000) to the original investment ($5,000), there would be $6,000 in the investment after one year. This process can be written as:\r\n<p style=\"padding-left: 120px;\"><strong>\u00a0 Accumulated Amount = Initial Investment + Interest Earned in One Year<\/strong><\/p>\r\n<p style=\"padding-left: 270px;\"><strong>A = P + PRT<\/strong><\/p>\r\nUsing algebra, we can factor out the P to get: <strong>A = P(1 + RT)<\/strong>\r\n\r\nThis shows us that we can also determine the amount of accumulated after one year by multiplying P by (1 + RT).\r\n<p style=\"padding-left: 150px;\">A = $5,000 (1 + (0.20)(1 year))<\/p>\r\n<p style=\"padding-left: 150px;\">A = $5,000 (1.20)<\/p>\r\nConverting 1.20 into a percent would give us 120%. Think of 120% as 100% + 20%, or the original plus the 20% increase. So, 1.20 represents a 20% increase per year.\r\n\r\nThe investment starts the second year with $6,000. We can use the same process to get the amount accumulated after the second year:\r\n<p style=\"padding-left: 120px;\">A = $6,000 (1.20)<\/p>\r\n<p style=\"padding-left: 120px;\">A = $7,200 (this is the amount accumulated after 2 years)<\/p>\r\nSetting up the pattern for multiple years, we have:\r\n<table style=\"border-collapse: collapse; width: 95.1518%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 13.411%; text-align: center;\">t = number of years<\/td>\r\n<td style=\"width: 61.8619%;\">Pattern<\/td>\r\n<td style=\"width: 29.4588%;\">y = Amount Accumulated<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 13.411%; text-align: center;\">0<\/td>\r\n<td style=\"width: 61.8619%;\">$5,000 (this is the initial investment)<\/td>\r\n<td style=\"width: 29.4588%;\">$5,000<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 13.411%; text-align: center;\">1<\/td>\r\n<td style=\"width: 61.8619%;\">$5,000 x (1.20)<\/td>\r\n<td style=\"width: 29.4588%;\">$6,000<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 13.411%; text-align: center;\">2<\/td>\r\n<td style=\"width: 61.8619%;\">$5,000 x (1.20)(1.20)\u00a0\u00a0\u00a0 (this is the accumulated amount at the end of the previous year multiplied by 1.20)<\/td>\r\n<td style=\"width: 29.4588%;\">$7,200<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 13.411%; text-align: center;\">3<\/td>\r\n<td style=\"width: 61.8619%;\">$5,000 x (1.20)(1.20)(1.20)<\/td>\r\n<td style=\"width: 29.4588%;\">$8,640<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 13.411%; text-align: center;\">4<\/td>\r\n<td style=\"width: 61.8619%;\">$5,000 x (1.20)(1.20)(1.20)(1.20)<\/td>\r\n<td style=\"width: 29.4588%;\">$10,368<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 13.411%; text-align: center;\">5<\/td>\r\n<td style=\"width: 61.8619%;\">$5,000 x (1.20)<sup>5<\/sup> \u00a0\u00a0\u00a0 (switch to notation using exponents)<\/td>\r\n<td style=\"width: 29.4588%;\">$12,441.60<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe general pattern is: <strong>y = $5,000(1.20)<\/strong><sup><strong>t<\/strong><\/sup>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Key Takeaways<\/h3>\r\nThe general equation for exponential growth is:\r\n<p style=\"text-align: center;\"><strong>y = a(b)<sup>t<\/sup><\/strong><\/p>\r\n\r\n<ul>\r\n \t<li>Where a is the initial amount<\/li>\r\n \t<li>b is (1 + <strong>growth rate<\/strong>) or (1 + r).\r\n<ul>\r\n \t<li>b is called the <strong>base.<\/strong><\/li>\r\n \t<li>b is sometimes called <strong>growth factor<\/strong>.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example<\/h3>\r\n<table style=\"border-collapse: collapse; width: 117.282%; height: 49px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 25px;\">\r\n<td style=\"width: 33.3333%; text-align: center; height: 25px;\"><strong>Initial Investment<\/strong><\/td>\r\n<td style=\"width: 33.3333%; text-align: center; height: 25px;\"><strong>Simple Interest Rate<\/strong><\/td>\r\n<td style=\"width: 50.6152%; text-align: center; height: 25px;\"><strong>Model, where y = amount accumulated and <\/strong><strong>t = number of years<\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 33.3333%; text-align: center; height: 12px;\">$300<\/td>\r\n<td style=\"width: 33.3333%; text-align: center; height: 12px;\">5%<\/td>\r\n<td style=\"width: 50.6152%; text-align: center; height: 12px;\">y = 300(1.05)<sup>t<\/sup><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 33.3333%; text-align: center; height: 12px;\">$1,000<\/td>\r\n<td style=\"width: 33.3333%; text-align: center; height: 12px;\">8.5%<\/td>\r\n<td style=\"width: 50.6152%; text-align: center; height: 12px;\">y = 1000(1.085)<sup>t<\/sup><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%; text-align: center;\">$25,000<\/td>\r\n<td style=\"width: 33.3333%; text-align: center;\">3%<\/td>\r\n<td style=\"width: 50.6152%; text-align: center;\">y = 25000(1.03)<sup>t<\/sup><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nThe exponential growth model (equation) can be made more generic by letting the initial investment (principle) be some other quantity.\r\n<div class=\"textbox exercises\">\r\n<h3>TRY IT<\/h3>\r\nFill in the missing cells based on the example in the first row.\r\n<table style=\"border-collapse: collapse; width: 100%; height: 73px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 25px;\">\r\n<td style=\"width: 25%; text-align: center; height: 25px;\"><strong>Initial Amount<\/strong><\/td>\r\n<td style=\"width: 25%; text-align: center; height: 25px;\"><strong>Growth Rate<\/strong><\/td>\r\n<td style=\"width: 25%; text-align: center; height: 25px;\"><strong>b = base (or growth factor)<\/strong><\/td>\r\n<td style=\"width: 25%; text-align: center; height: 25px;\"><strong>y = amount accumulated, t = time unit<\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 25%; text-align: center; height: 12px;\">30<\/td>\r\n<td style=\"width: 25%; text-align: center; height: 12px;\">1%<\/td>\r\n<td style=\"width: 25%; text-align: center; height: 12px;\">1.01<\/td>\r\n<td style=\"width: 25%; text-align: center; height: 12px;\">y = 30 (1.01)<sup>t<\/sup><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 25%; text-align: center; height: 12px;\">150<\/td>\r\n<td style=\"width: 25%; text-align: center; height: 12px;\">[reveal-answer q=\"223328\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"223328\"]35%[\/hidden-answer]<\/td>\r\n<td style=\"width: 25%; text-align: center; height: 12px;\">1.35<\/td>\r\n<td style=\"width: 25%; text-align: center; height: 12px;\">y = 150 (1.35)<sup>t<\/sup><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 25%; text-align: center; height: 12px;\">[reveal-answer q=\"896337\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"896337\"]450[\/hidden-answer]<\/td>\r\n<td style=\"width: 25%; text-align: center; height: 12px;\">99%<\/td>\r\n<td style=\"width: 25%; text-align: center; height: 12px;\">[reveal-answer q=\"684268\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"684268\"]1.99[\/hidden-answer]<\/td>\r\n<td style=\"width: 25%; text-align: center; height: 12px;\">y = 450 (1.99)<sup>t<\/sup><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 25%; text-align: center; height: 12px;\">[reveal-answer q=\"247460\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"247460\"]5[\/hidden-answer]<\/td>\r\n<td style=\"width: 25%; text-align: center; height: 12px;\">100%<\/td>\r\n<td style=\"width: 25%; text-align: center; height: 12px;\">[reveal-answer q=\"540341\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"540341\"]2[\/hidden-answer]<\/td>\r\n<td style=\"width: 25%; text-align: center; height: 12px;\">y = 5 (2)<sup>t<\/sup><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%; text-align: center;\">900<\/td>\r\n<td style=\"width: 25%; text-align: center;\">[reveal-answer q=\"106261\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"106261\"]200%[\/hidden-answer]<\/td>\r\n<td style=\"width: 25%; text-align: center;\">3<\/td>\r\n<td style=\"width: 25%; text-align: center;\">y = 900 (3)<sup>t<\/sup><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%; text-align: center;\">6<\/td>\r\n<td style=\"width: 25%; text-align: center;\">300%<\/td>\r\n<td style=\"width: 25%; text-align: center;\">[reveal-answer q=\"328397\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"328397\"]4[\/hidden-answer]<\/td>\r\n<td style=\"width: 25%; text-align: center;\">y = 6 (4)<sup>t<\/sup><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<p style=\"text-align: center;\"><span style=\"color: #ff0000;\"><strong>This is the end of the section. Close this tab and proceed to the corresponding assignment.<\/strong><\/span><\/p>\r\n&nbsp;","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Apply and construct exponential models using the form y = a(b)<sup>x <\/sup><\/li>\n<\/ul>\n<\/div>\n<p>Just as in linear growth, we can model quantities that grow in a non-linear pattern. Consider exponential growth.<\/p>\n<p style=\"text-align: center;\"><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/02\/05224552\/pennies.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-5610\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/02\/05224552\/pennies-300x198.jpg\" alt=\"\" width=\"486\" height=\"321\" \/><\/a><\/p>\n<h3>Exponential Models<\/h3>\n<p>Consider the older brother who offered to pay his younger brother 1 penny per day to help him mow lawns and rake leaves. The younger brother said, &#8220;No thanks! But, I will work for you for 30 days in a row. You will pay me 1 penny on the first day and then each day after that you will double that amount you paid me the previous day.&#8221;<\/p>\n<p>Let&#8217;s model that situation with a table and a graph. Let x represent the number of days, with zero representing Day 1. Let y = the amount of money the younger brother earns in this model.<\/p>\n<table style=\"width: 381px;\">\n<tbody>\n<tr>\n<td style=\"width: 67.75px; text-align: left;\"><strong>Day<\/strong><\/td>\n<td style=\"width: 227.15px;\"><strong>x = number of days (Day 1 is 0)<\/strong><\/td>\n<td style=\"width: 165.3px;\"><strong>y = amount of money<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 67.75px;\">1<\/td>\n<td style=\"width: 227.15px;\">0<\/td>\n<td style=\"width: 165.3px;\">$0.01<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 67.75px;\">2<\/td>\n<td style=\"width: 227.15px;\">1<\/td>\n<td style=\"width: 165.3px;\">$0.02<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 67.75px;\">3<\/td>\n<td style=\"width: 227.15px;\">2<\/td>\n<td style=\"width: 165.3px;\">$0.04<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 67.75px;\">4<\/td>\n<td style=\"width: 227.15px;\">3<\/td>\n<td style=\"width: 165.3px;\">$0.08<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 67.75px;\">5<\/td>\n<td style=\"width: 227.15px;\">4<\/td>\n<td style=\"width: 165.3px;\">$0.16<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 67.75px;\">10<\/td>\n<td style=\"width: 227.15px;\">9<\/td>\n<td style=\"width: 165.3px;\">$10.24<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 67.75px;\">25<\/td>\n<td style=\"width: 227.15px;\">24<\/td>\n<td style=\"width: 165.3px;\">$167,772.16<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 67.75px;\">26<\/td>\n<td style=\"width: 227.15px;\">25<\/td>\n<td style=\"width: 165.3px;\">$333,544<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 67.75px;\">27<\/td>\n<td style=\"width: 227.15px;\">26<\/td>\n<td style=\"width: 165.3px;\">$671,088.64<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 67.75px;\">28<\/td>\n<td style=\"width: 227.15px;\">27<\/td>\n<td style=\"width: 165.3px;\">$1,342,177.28<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 67.75px;\">29<\/td>\n<td style=\"width: 227.15px;\">28<\/td>\n<td style=\"width: 165.3px;\">$2,684,354.56<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 67.75px;\">30<\/td>\n<td style=\"width: 227.15px;\">29<\/td>\n<td style=\"width: 165.3px;\">$5,368,709.12<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/02\/06213343\/Expo-1.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-5614 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/02\/06213343\/Expo-1.jpg\" alt=\"\" width=\"679\" height=\"521\" \/><\/a><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>Notice the table of x- and y-values, along with the graph. Younger brother would only be making $10.24 on Day 10. But, then something happens around Day 25. The pay drastically increases. By Day 30, he would be making over $5 million.<\/p>\n<p>Graphs shaped like this are modeling <strong>exponential growth<\/strong>. Let&#8217;s examine the table to determine the equation that models the younger brother&#8217;s pay.<\/p>\n<table style=\"width: 979px; height: 196px;\">\n<tbody>\n<tr style=\"height: 28px;\">\n<td style=\"width: 51.8667px; height: 28px;\"><strong>Day<\/strong><\/td>\n<td style=\"width: 120.75px; height: 28px;\"><strong>x = number of days (Day 1 is 0)<\/strong><\/td>\n<td style=\"width: 510.533px; height: 28px;\"><strong>Pattern<\/strong><\/td>\n<td style=\"width: 245.133px; height: 28px;\"><strong>y = amount of money<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 51.8667px; height: 14px;\">1<\/td>\n<td style=\"width: 120.75px; height: 14px;\">0<\/td>\n<td style=\"width: 510.533px; height: 14px;\">Pay on Day 1 = $0.01<\/td>\n<td style=\"width: 245.133px; height: 14px;\">$0.01<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 51.8667px; height: 14px;\">2<\/td>\n<td style=\"width: 120.75px; height: 14px;\">1<\/td>\n<td style=\"width: 510.533px; height: 14px;\">Pay on Day 2 = $0.01 x 2 \u00a0 \u00a0 (When something doubles, that means multiply by 2.)<\/td>\n<td style=\"width: 245.133px; height: 14px;\">$0.02<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 51.8667px; height: 14px;\">3<\/td>\n<td style=\"width: 120.75px; height: 14px;\">2<\/td>\n<td style=\"width: 510.533px; height: 14px;\">Pay on Day 3 = $0.01 x 2 x 2<\/td>\n<td style=\"width: 245.133px; height: 14px;\">$0.04<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 51.8667px; height: 14px;\">4<\/td>\n<td style=\"width: 120.75px; height: 14px;\">3<\/td>\n<td style=\"width: 510.533px; height: 14px;\">Pay on Day 4 = $0.01 x 2<sup>3<\/sup> \u00a0\u00a0 (Switch notation to using exponents.)<\/td>\n<td style=\"width: 245.133px; height: 14px;\">$0.08<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 51.8667px; height: 14px;\">5<\/td>\n<td style=\"width: 120.75px; height: 14px;\">4<\/td>\n<td style=\"width: 510.533px; height: 14px;\">Pay on Day 5 = $0.01 x 2<sup>4<\/sup><\/td>\n<td style=\"width: 245.133px; height: 14px;\">$0.16<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 51.8667px; height: 14px;\">10<\/td>\n<td style=\"width: 120.75px; height: 14px;\">9<\/td>\n<td style=\"width: 510.533px; height: 14px;\">Pay on Day 10 = $0.01 x 2<sup>9<\/sup><\/td>\n<td style=\"width: 245.133px; height: 14px;\">$10.24<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 51.8667px; height: 14px;\">25<\/td>\n<td style=\"width: 120.75px; height: 14px;\">24<\/td>\n<td style=\"width: 510.533px; height: 14px;\">Pay on Day 25 = $0.01 x 2<sup>24<\/sup><\/td>\n<td style=\"width: 245.133px; height: 14px;\">$167,772.16<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 51.8667px; height: 14px;\">26<\/td>\n<td style=\"width: 120.75px; height: 14px;\">25<\/td>\n<td style=\"width: 510.533px; height: 14px;\">Pay on Day 26 = $0.01 x 2<sup>25<\/sup><\/td>\n<td style=\"width: 245.133px; height: 14px;\">$333,544<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 51.8667px; height: 14px;\">27<\/td>\n<td style=\"width: 120.75px; height: 14px;\">26<\/td>\n<td style=\"width: 510.533px; height: 14px;\">Pay on Day 27 = $0.01 x 2<sup>26<\/sup><\/td>\n<td style=\"width: 245.133px; height: 14px;\">$671,088.64<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 51.8667px; height: 14px;\">28<\/td>\n<td style=\"width: 120.75px; height: 14px;\">27<\/td>\n<td style=\"width: 510.533px; height: 14px;\">Pay on Day 28 = $0.01 x 2<sup>27<\/sup><\/td>\n<td style=\"width: 245.133px; height: 14px;\">$1,342,177.28<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 51.8667px; height: 14px;\">29<\/td>\n<td style=\"width: 120.75px; height: 14px;\">28<\/td>\n<td style=\"width: 510.533px; height: 14px;\">Pay on Day 29 = $0.01 x 2<sup>28<\/sup><\/td>\n<td style=\"width: 245.133px; height: 14px;\">$2,684,354.56<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 51.8667px; height: 14px;\">30<\/td>\n<td style=\"width: 120.75px; height: 14px;\">29<\/td>\n<td style=\"width: 510.533px; height: 14px;\">Pay on Day 30 = $0.01 x 2<sup>29<\/sup><\/td>\n<td style=\"width: 245.133px; height: 14px;\">$5,368,709.12<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Letting x be the number of days after the first day, then Day 1 is 0 and Day 2 is 1, and so on. Then letting y be the amount of pay younger brother is going to receive, the last day of the month younger brother receives $0.01 x 2<sup>29<\/sup>. Now, letting x represent any day, the equation becomes: <strong>y = $0.01(2)<sup>x <\/sup><\/strong>.<\/p>\n<p>Suppose that the younger brother said the pay would be <strong>triple<\/strong> the amount paid on the previous day. The model would be:<\/p>\n<p style=\"text-align: center;\"><strong>y = $0.01(3)<sup>x<\/sup><\/strong><\/p>\n<p>If the younger brother had used the word <strong>quadruple<\/strong>, the model would be:<\/p>\n<p style=\"text-align: center;\"><strong>y = $0.01(4)<sup>x<\/sup><\/strong><\/p>\n<h3>Investing Money<\/h3>\n<p>Consider the scenario of investing money.<\/p>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/02\/07000134\/money.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-5622 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/02\/07000134\/money-300x203.jpg\" alt=\"\" width=\"476\" height=\"322\" \/><\/a><\/p>\n<p>Suppose $5,000 is invested into an investment that earns 20% simple interest per year. To compute the interest earned in the first year, use the simple interest formula: <strong>Interest = Principle x Rate x Time (in years), <\/strong>or<strong> I = PRT.<\/strong><\/p>\n<p>After one year, the interest earned would be:<\/p>\n<p style=\"padding-left: 90px;\">I = PRT<\/p>\n<p style=\"padding-left: 90px;\">\u00a0 = ($5,000) x (0.20) x (1 year)<\/p>\n<p style=\"padding-left: 90px;\">\u00a0 = $1,000<\/p>\n<p>Then, adding the interest earned in the first year ($1,000) to the original investment ($5,000), there would be $6,000 in the investment after one year. This process can be written as:<\/p>\n<p style=\"padding-left: 120px;\"><strong>\u00a0 Accumulated Amount = Initial Investment + Interest Earned in One Year<\/strong><\/p>\n<p style=\"padding-left: 270px;\"><strong>A = P + PRT<\/strong><\/p>\n<p>Using algebra, we can factor out the P to get: <strong>A = P(1 + RT)<\/strong><\/p>\n<p>This shows us that we can also determine the amount of accumulated after one year by multiplying P by (1 + RT).<\/p>\n<p style=\"padding-left: 150px;\">A = $5,000 (1 + (0.20)(1 year))<\/p>\n<p style=\"padding-left: 150px;\">A = $5,000 (1.20)<\/p>\n<p>Converting 1.20 into a percent would give us 120%. Think of 120% as 100% + 20%, or the original plus the 20% increase. So, 1.20 represents a 20% increase per year.<\/p>\n<p>The investment starts the second year with $6,000. We can use the same process to get the amount accumulated after the second year:<\/p>\n<p style=\"padding-left: 120px;\">A = $6,000 (1.20)<\/p>\n<p style=\"padding-left: 120px;\">A = $7,200 (this is the amount accumulated after 2 years)<\/p>\n<p>Setting up the pattern for multiple years, we have:<\/p>\n<table style=\"border-collapse: collapse; width: 95.1518%;\">\n<tbody>\n<tr>\n<td style=\"width: 13.411%; text-align: center;\">t = number of years<\/td>\n<td style=\"width: 61.8619%;\">Pattern<\/td>\n<td style=\"width: 29.4588%;\">y = Amount Accumulated<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 13.411%; text-align: center;\">0<\/td>\n<td style=\"width: 61.8619%;\">$5,000 (this is the initial investment)<\/td>\n<td style=\"width: 29.4588%;\">$5,000<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 13.411%; text-align: center;\">1<\/td>\n<td style=\"width: 61.8619%;\">$5,000 x (1.20)<\/td>\n<td style=\"width: 29.4588%;\">$6,000<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 13.411%; text-align: center;\">2<\/td>\n<td style=\"width: 61.8619%;\">$5,000 x (1.20)(1.20)\u00a0\u00a0\u00a0 (this is the accumulated amount at the end of the previous year multiplied by 1.20)<\/td>\n<td style=\"width: 29.4588%;\">$7,200<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 13.411%; text-align: center;\">3<\/td>\n<td style=\"width: 61.8619%;\">$5,000 x (1.20)(1.20)(1.20)<\/td>\n<td style=\"width: 29.4588%;\">$8,640<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 13.411%; text-align: center;\">4<\/td>\n<td style=\"width: 61.8619%;\">$5,000 x (1.20)(1.20)(1.20)(1.20)<\/td>\n<td style=\"width: 29.4588%;\">$10,368<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 13.411%; text-align: center;\">5<\/td>\n<td style=\"width: 61.8619%;\">$5,000 x (1.20)<sup>5<\/sup> \u00a0\u00a0\u00a0 (switch to notation using exponents)<\/td>\n<td style=\"width: 29.4588%;\">$12,441.60<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The general pattern is: <strong>y = $5,000(1.20)<\/strong><sup><strong>t<\/strong><\/sup><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Key Takeaways<\/h3>\n<p>The general equation for exponential growth is:<\/p>\n<p style=\"text-align: center;\"><strong>y = a(b)<sup>t<\/sup><\/strong><\/p>\n<ul>\n<li>Where a is the initial amount<\/li>\n<li>b is (1 + <strong>growth rate<\/strong>) or (1 + r).\n<ul>\n<li>b is called the <strong>base.<\/strong><\/li>\n<li>b is sometimes called <strong>growth factor<\/strong>.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example<\/h3>\n<table style=\"border-collapse: collapse; width: 117.282%; height: 49px;\">\n<tbody>\n<tr style=\"height: 25px;\">\n<td style=\"width: 33.3333%; text-align: center; height: 25px;\"><strong>Initial Investment<\/strong><\/td>\n<td style=\"width: 33.3333%; text-align: center; height: 25px;\"><strong>Simple Interest Rate<\/strong><\/td>\n<td style=\"width: 50.6152%; text-align: center; height: 25px;\"><strong>Model, where y = amount accumulated and <\/strong><strong>t = number of years<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 33.3333%; text-align: center; height: 12px;\">$300<\/td>\n<td style=\"width: 33.3333%; text-align: center; height: 12px;\">5%<\/td>\n<td style=\"width: 50.6152%; text-align: center; height: 12px;\">y = 300(1.05)<sup>t<\/sup><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 33.3333%; text-align: center; height: 12px;\">$1,000<\/td>\n<td style=\"width: 33.3333%; text-align: center; height: 12px;\">8.5%<\/td>\n<td style=\"width: 50.6152%; text-align: center; height: 12px;\">y = 1000(1.085)<sup>t<\/sup><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%; text-align: center;\">$25,000<\/td>\n<td style=\"width: 33.3333%; text-align: center;\">3%<\/td>\n<td style=\"width: 50.6152%; text-align: center;\">y = 25000(1.03)<sup>t<\/sup><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>The exponential growth model (equation) can be made more generic by letting the initial investment (principle) be some other quantity.<\/p>\n<div class=\"textbox exercises\">\n<h3>TRY IT<\/h3>\n<p>Fill in the missing cells based on the example in the first row.<\/p>\n<table style=\"border-collapse: collapse; width: 100%; height: 73px;\">\n<tbody>\n<tr style=\"height: 25px;\">\n<td style=\"width: 25%; text-align: center; height: 25px;\"><strong>Initial Amount<\/strong><\/td>\n<td style=\"width: 25%; text-align: center; height: 25px;\"><strong>Growth Rate<\/strong><\/td>\n<td style=\"width: 25%; text-align: center; height: 25px;\"><strong>b = base (or growth factor)<\/strong><\/td>\n<td style=\"width: 25%; text-align: center; height: 25px;\"><strong>y = amount accumulated, t = time unit<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 25%; text-align: center; height: 12px;\">30<\/td>\n<td style=\"width: 25%; text-align: center; height: 12px;\">1%<\/td>\n<td style=\"width: 25%; text-align: center; height: 12px;\">1.01<\/td>\n<td style=\"width: 25%; text-align: center; height: 12px;\">y = 30 (1.01)<sup>t<\/sup><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 25%; text-align: center; height: 12px;\">150<\/td>\n<td style=\"width: 25%; text-align: center; height: 12px;\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q223328\">Show Answer<\/span><\/p>\n<div id=\"q223328\" class=\"hidden-answer\" style=\"display: none\">35%<\/div>\n<\/div>\n<\/td>\n<td style=\"width: 25%; text-align: center; height: 12px;\">1.35<\/td>\n<td style=\"width: 25%; text-align: center; height: 12px;\">y = 150 (1.35)<sup>t<\/sup><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 25%; text-align: center; height: 12px;\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q896337\">Show Answer<\/span><\/p>\n<div id=\"q896337\" class=\"hidden-answer\" style=\"display: none\">450<\/div>\n<\/div>\n<\/td>\n<td style=\"width: 25%; text-align: center; height: 12px;\">99%<\/td>\n<td style=\"width: 25%; text-align: center; height: 12px;\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q684268\">Show Answer<\/span><\/p>\n<div id=\"q684268\" class=\"hidden-answer\" style=\"display: none\">1.99<\/div>\n<\/div>\n<\/td>\n<td style=\"width: 25%; text-align: center; height: 12px;\">y = 450 (1.99)<sup>t<\/sup><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 25%; text-align: center; height: 12px;\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q247460\">Show Answer<\/span><\/p>\n<div id=\"q247460\" class=\"hidden-answer\" style=\"display: none\">5<\/div>\n<\/div>\n<\/td>\n<td style=\"width: 25%; text-align: center; height: 12px;\">100%<\/td>\n<td style=\"width: 25%; text-align: center; height: 12px;\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q540341\">Show Answer<\/span><\/p>\n<div id=\"q540341\" class=\"hidden-answer\" style=\"display: none\">2<\/div>\n<\/div>\n<\/td>\n<td style=\"width: 25%; text-align: center; height: 12px;\">y = 5 (2)<sup>t<\/sup><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25%; text-align: center;\">900<\/td>\n<td style=\"width: 25%; text-align: center;\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q106261\">Show Answer<\/span><\/p>\n<div id=\"q106261\" class=\"hidden-answer\" style=\"display: none\">200%<\/div>\n<\/div>\n<\/td>\n<td style=\"width: 25%; text-align: center;\">3<\/td>\n<td style=\"width: 25%; text-align: center;\">y = 900 (3)<sup>t<\/sup><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25%; text-align: center;\">6<\/td>\n<td style=\"width: 25%; text-align: center;\">300%<\/td>\n<td style=\"width: 25%; text-align: center;\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q328397\">Show Answer<\/span><\/p>\n<div id=\"q328397\" class=\"hidden-answer\" style=\"display: none\">4<\/div>\n<\/div>\n<\/td>\n<td style=\"width: 25%; text-align: center;\">y = 6 (4)<sup>t<\/sup><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p style=\"text-align: center;\"><span style=\"color: #ff0000;\"><strong>This is the end of the section. 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