{"id":417,"date":"2022-04-13T10:56:05","date_gmt":"2022-04-13T10:56:05","guid":{"rendered":"http:\/\/10thclass.deltapublications.in\/?page_id=417"},"modified":"2025-03-20T08:18:16","modified_gmt":"2025-03-20T08:18:16","slug":"v-4-exponential-growth-and-decay-word-problems","status":"publish","type":"page","link":"https:\/\/10thclass.deltapublications.in\/index.php\/v-4-exponential-growth-and-decay-word-problems\/","title":{"rendered":"V.4 Exponential growth and decay: word problems"},"content":{"rendered":"\n<h2 class=\"wp-block-heading has-text-align-center has-text-color\" style=\"color:#00056d;text-transform:uppercase\"><strong> Exponential growth and decay: word problems<\/strong><\/h2>\n\n\n\n<p class=\"has-text-color has-link-color has-huge-font-size wp-elements-203290aba75681a4fb07a0852f132b50\" style=\"color:#74008b;text-transform:capitalize\">key notes :<\/p>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-b4a8515d5ad9b38f8eb208c23ff47fe6\" style=\"color:#000060\">\ud83c\udfaf <strong>Understanding Exponential Growth and Decay<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li><strong>Exponential growth<\/strong> happens when a quantity increases by a constant <strong>percentage<\/strong> over equal time periods.<\/li>\n\n\n\n<li><strong>Exponential decay<\/strong> happens when a quantity decreases by a constant <strong>percentage<\/strong> over equal time periods.<\/li>\n\n\n\n<li>The general formula for both is:<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-align-center has-large-font-size\">A = A<sub>0<\/sub> \u00d7 (1\u00b1r)t<\/p>\n\n\n\n<p class=\"has-large-font-size\"><strong>Where<\/strong>:<\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>A = final amount<\/li>\n\n\n\n<li>A<sub>0<\/sub>\u200b = initial amount<\/li>\n\n\n\n<li>r = rate of increase or decrease (as a decimal)<\/li>\n\n\n\n<li>t = time (usually in years, but can be other units)<\/li>\n\n\n\n<li><strong>+<\/strong> for growth, <strong>&#8211;<\/strong> for decay<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading has-text-color has-link-color has-large-font-size wp-elements-cbfe21bca5aa42caf99daa51c39afdab\" style=\"color:#000060\">\ud83d\udd0d<strong>Key Characteristics<\/strong><\/h3>\n\n\n\n<p class=\"has-large-font-size\">\u2705 <strong>Exponential Growth:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>The value increases rapidly after a certain point (e.g., population growth, investments).<\/li>\n\n\n\n<li>Graph rises steeply.<\/li>\n\n\n\n<li>Example: Bacteria doubling every hour.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-large-font-size\">\u2705 <strong>Exponential Decay:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>The value decreases but never quite hits zero (e.g., radioactive decay, depreciation).<\/li>\n\n\n\n<li>Graph falls but flattens over time.<\/li>\n\n\n\n<li>Example: Half-life of a substance.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading has-text-color has-link-color has-large-font-size wp-elements-655a99c535bc84231674bcac7a7a4426\" style=\"color:#000060\">\ud83d\udee0\ufe0f<strong>Steps to Solve Word Problems<\/strong><\/h3>\n\n\n\n<p class=\"has-large-font-size\">1\ufe0f\u20e3 <strong>Identify the type<\/strong> \u2014 Is it growth (+) or decay (-)?<br>2\ufe0f\u20e3 <strong>Identify the variables<\/strong> \u2014 What are the initial amount, rate, and time?<br>3\ufe0f\u20e3 <strong>Set up the formula<\/strong> \u2014 Plug the values into the formula correctly.<br>4\ufe0f\u20e3 <strong>Solve carefully<\/strong> \u2014 Be cautious with powers and decimals.<br>5\ufe0f\u20e3 <strong>Check the answer<\/strong> \u2014 Does it make sense? (e.g., a population can\u2019t be negative!)<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading has-text-color has-link-color has-large-font-size wp-elements-9d22f9803b149ce5fd3f19cffe9624f4\" style=\"color:#000060\">\ud83d\udd25<strong>Example Word Problems<\/strong><\/h3>\n\n\n\n<p class=\"has-large-font-size\">\u2705 <strong>Example 1: Population Growth<\/strong><\/p>\n\n\n\n<p class=\"has-large-font-size\">A town\u2019s population is 5,000 and grows at a rate of 2% per year. What will the population be in 10 years? <\/p>\n\n\n\n<p class=\"has-large-font-size\">A = 5000 \u00d7 (1+0.02)<sup>10<\/sup><\/p>\n\n\n\n<p class=\"has-large-font-size\">A = 5000 \u00d7 (1+0.02)<sup>10 <\/sup><\/p>\n\n\n\n<p class=\"has-large-font-size\">A \u2248 5000 \u00d7 1.219 = 6095<\/p>\n\n\n\n<p class=\"has-large-font-size\">\ud83d\udc49 <strong>Answer:<\/strong> The population will be about <strong>6,095 people<\/strong>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-60d8c3aed1769c59e64510aa89b0a47d\" style=\"color:#000060\">\u2705 <strong>Example 2: Radioactive Decay<\/strong><\/p>\n\n\n\n<p class=\"has-large-font-size\">A radioactive substance has a mass of 120 grams and decays at a rate of 5% per hour. How much is left after 8 hours? <\/p>\n\n\n\n<p class=\"has-large-font-size\">A = 120 \u00d7 (1\u22120.05)<sup>8<\/sup><\/p>\n\n\n\n<p class=\"has-large-font-size\">A = 120 \u00d7 (0.95)<sup>8<\/sup><\/p>\n\n\n\n<p class=\"has-large-font-size\">A \u2248 120\u00d70.663 = 79.56\u2009grams<\/p>\n\n\n\n<p class=\"has-large-font-size\">\ud83d\udc49 <strong>Answer:<\/strong> About <strong>79.56 grams<\/strong> remain.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-5e42eab9f56a4c7c65b7ca29c9ce3ddd\" style=\"color:#000060\">\u2705 <strong>Example 3: Investment Growth<\/strong><\/p>\n\n\n\n<p class=\"has-large-font-size\">A $1,000 investment grows at 6% annually. How much will it be worth after 5 years? <\/p>\n\n\n\n<p class=\"has-large-font-size\">A = 1000 \u00d7 (1+0.06)<sup>5<\/sup><\/p>\n\n\n\n<p class=\"has-large-font-size\">A = 1000 \u00d7 (1.06)<sup>5<\/sup><\/p>\n\n\n\n<p class=\"has-large-font-size\">A \u2248 1000 \u00d7 1.338=1338<\/p>\n\n\n\n<p class=\"has-large-font-size\">\ud83d\udc49 <strong>Answer:<\/strong> The investment will grow to <strong>$1,338<\/strong>.<\/p>\n\n\n\n<p class=\"has-text-align-center has-text-color has-large-font-size\" style=\"color:#105000\"><strong>Learn with an example<\/strong><\/p>\n\n\n\n<div class=\"wp-block-group has-background has-large-font-size\" style=\"background-color:#def7ac\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p class=\"has-background-background-color has-text-color has-background has-link-color wp-elements-eb59ecd680b617755480761c07c5386d\" style=\"color:#b00012\"><strong> \ud83d\uddfc A city&#8217;s population is currently 150,000. If the population doubles every 30 years, what will the population be 60 years from now?<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>First, find out how many times the population will double. Divide the number of years by how long it takes for the population to double.<\/li>\n\n\n\n<li>60 \u00f7 30 = 2<\/li>\n\n\n\n<li>The population will double 2 times.<\/li>\n\n\n\n<li>Now figure out what the population will be after it doubles 2 times. Multiply the population by 2 a total of 2 times.<\/li>\n\n\n\n<li>150,000 \u00d7 2 \u00d7 2 = 600,000<\/li>\n\n\n\n<li>That calculation could also be written with exponents:<\/li>\n\n\n\n<li>150,000 \u00d7 2<sup>2<\/sup> = 600,000<\/li>\n\n\n\n<li>After 60 years, the population will be 600,000 people.<\/li>\n<\/ul>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-background has-large-font-size\" style=\"background-color:#f9bdbd\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<div class=\"wp-block-group has-background-background-color has-background\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p class=\"has-text-color has-link-color wp-elements-25432c74c391dbc99978ddb4be5dd9f4\" style=\"color:#b00012\"><strong> \ud83d\uddfc A virus is spreading around Westminster. At the moment, 490 people are infected. If the virus is spreading at a rate of 15% each day, how many people in all will have caught the virus in 14 days?<\/strong><\/p>\n\n\n\n<p><em>If necessary, round your answer to the nearest whole number.<\/em><\/p>\n\n\n\n<p>&nbsp;_________people<\/p>\n<\/div><\/div>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Plug in 490 people for the initial amount, 0.15 for the rate of increase, and 14 days for the time elapsed.<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td><em>y<\/em><\/td><td>&nbsp;=&nbsp;<\/td><td><em>a<\/em>(1 +&nbsp;<em>r<\/em>)<sup><em>t<\/em><\/sup><\/td><td><em><\/em><\/td><\/tr><tr><td><\/td><\/tr><tr><td><em>y<\/em><\/td><td>&nbsp;=&nbsp;<\/td><td>490(1 + 0.15)<sup>14<\/sup><\/td><td><em>Plug in<\/em><\/td><\/tr><tr><td><\/td><\/tr><tr><td><em>y<\/em><\/td><td>&nbsp;=&nbsp;<\/td><td>490(1.15)<sup>14<\/sup><\/td><td><em>Add<\/em><\/td><\/tr><tr><td><\/td><\/tr><tr><td><em>y<\/em><\/td><td>&nbsp;\u2248&nbsp;<\/td><td>490(7.075)<\/td><td><em>Simplify<\/em><\/td><\/tr><tr><td><\/td><\/tr><tr><td><em>y<\/em><\/td><td>&nbsp;\u2248&nbsp;<\/td><td>3,466.75<\/td><td><em>Multiply<\/em><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Round to the nearest whole number.<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td>3,466.75<\/td><td>&nbsp;\u2192&nbsp;<\/td><td>3,467<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<ul class=\"wp-block-list\">\n<li>To the nearest whole number, 3,467 people will have caught the virus.<\/li>\n<\/ul>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-background has-large-font-size\" style=\"background-color:#c4edf9\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<div class=\"wp-block-group has-background-background-color has-background\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p class=\"has-text-color has-link-color wp-elements-9a34236405ae3bc45edaa3db02aad6bd\" style=\"color:#b00012\"><strong>   \ud83d\uddfc Sam has put&nbsp;\u20b91,800,000 into a pension. The fund has an estimated annual return of 5%. If Sam doesn&#8217;t add any more money, how much can he expect to have in the fund in 20 years?<\/strong><\/p>\n\n\n\n<p><em>If necessary, round your answer to the nearest paisa.<\/em><\/p>\n\n\n\n<p>\u20b9________<br><\/p>\n<\/div><\/div>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Plug in&nbsp;\u20b91,800,000 for the initial amount, 0.05 for the rate of increase, and 20 years for the time elapsed.<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td><em>y<\/em><\/td><td>&nbsp;=&nbsp;<\/td><td><em>a<\/em>(1 +&nbsp;<em>r<\/em>)<sup><em>t<\/em><\/sup><\/td><td><em><\/em><\/td><\/tr><tr><td><\/td><\/tr><tr><td><em>y<\/em><\/td><td>&nbsp;=&nbsp;<\/td><td>1,800,000(1 + 0.05)<sup>20<\/sup><\/td><td><em>Plug in<\/em><\/td><\/tr><tr><td><\/td><\/tr><tr><td><em>y<\/em><\/td><td>&nbsp;=&nbsp;<\/td><td>1,800,000(1.05)<sup>20<\/sup><\/td><td><em>Add<\/em><\/td><\/tr><tr><td><\/td><\/tr><tr><td><em>y<\/em><\/td><td>&nbsp;\u2248&nbsp;<\/td><td>1,800,000(2.653297705)<\/td><td><em>Simplify<\/em><\/td><\/tr><tr><td><\/td><\/tr><tr><td><em>y<\/em><\/td><td>&nbsp;\u2248&nbsp;<\/td><td>4,775,935.869<\/td><td><em>Multiply<\/em><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Round to the nearest paisa.<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td>4,775,935.869<\/td><td>&nbsp;\u2192&nbsp;<\/td><td>4,775,935.87<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<ul class=\"wp-block-list\">\n<li>To the nearest paisa, Sam can expect to have&nbsp;\u20b94,775,935.87 in his pension.<\/li>\n<\/ul>\n<\/div><\/div>\n<\/div><\/div>\n\n\n\n<p class=\"has-text-color has-large-font-size\" style=\"color:#d90000\">Let&#8217;s practice!\ud83d\udd8a\ufe0f<\/p>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<figure class=\"wp-block-image size-full\"><a href=\"https:\/\/wordwall.net\/play\/88919\/956\/284\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-150.png\" alt=\"\" class=\"wp-image-7449\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-150.png 500w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-150-300x300.png 300w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-150-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><\/figure>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<figure class=\"wp-block-image size-full\"><a href=\"https:\/\/wordwall.net\/play\/73791\/556\/901\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-169.png\" alt=\"\" class=\"wp-image-7450\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-169.png 500w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-169-300x300.png 300w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-169-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><\/figure>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Exponential growth and decay: word problems key notes : \ud83c\udfaf Understanding Exponential Growth and Decay A = A0 \u00d7 (1\u00b1r)t Where: \ud83d\udd0dKey Characteristics \u2705 Exponential Growth: \u2705 Exponential Decay: \ud83d\udee0\ufe0fSteps to Solve Word Problems 1\ufe0f\u20e3 Identify the type \u2014 Is it growth (+) or decay (-)?2\ufe0f\u20e3 Identify the variables \u2014 What are the initial amount,<a class=\"more-link\" href=\"https:\/\/10thclass.deltapublications.in\/index.php\/v-4-exponential-growth-and-decay-word-problems\/\">Continue reading <span class=\"screen-reader-text\">&#8220;V.4 Exponential growth and decay: word problems&#8221;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"class_list":["post-417","page","type-page","status-publish","hentry","entry"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/417","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/comments?post=417"}],"version-history":[{"count":15,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/417\/revisions"}],"predecessor-version":[{"id":17656,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/417\/revisions\/17656"}],"wp:attachment":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/media?parent=417"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}