{"id":243465,"date":"2025-07-04T14:04:41","date_gmt":"2025-07-04T14:04:41","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=243465"},"modified":"2025-07-04T14:04:43","modified_gmt":"2025-07-04T14:04:43","slug":"the-population-of-a-colony-of-mosquitoes-obeys-the-law-of-uninhibited-growth","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/04\/the-population-of-a-colony-of-mosquitoes-obeys-the-law-of-uninhibited-growth\/","title":{"rendered":"The population of a colony of mosquitoes obeys the law of uninhibited growth"},"content":{"rendered":"\n<p>The population of a colony of mosquitoes obeys the law of uninhibited growth: Use this information to answer parts (a) through (c): (a) If N is the population of the colony and t is the time in days, express N as a function of t. Consider No is the original amount at t = 0 and k # 0 is a constant that represents the growth rate. N(t) = (Type an expression using t as the variable and in terms of e.) (b) The population of a colony of mosquitoes obeys the law of uninhibited growth. If there are 1000 mosquitoes initially and there are 1500 after day; what is the size of the colony after 4 days? Approximately mosquitoes (Do not round until the final answer: Then round to the nearest whole number as needed ) (c) How long is it until there are 40,000 mosquitoes? About days (Do not round until the final answer: Then round to the nearest tenth as needed )<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Part (a): Express N(t)N(t)N(t) as a function of ttt<\/h3>\n\n\n\n<p>The law of uninhibited growth for a population N(t)N(t)N(t) is modeled by the exponential growth equation:N(t)=N0ektN(t) = N_0 e^{kt}N(t)=N0\u200bekt<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>N(t)N(t)N(t) is the population at time ttt,<\/li>\n\n\n\n<li>N0N_0N0\u200b is the initial population at t=0t = 0t=0,<\/li>\n\n\n\n<li>kkk is the growth rate constant,<\/li>\n\n\n\n<li>eee is the base of the natural logarithm.<\/li>\n<\/ul>\n\n\n\n<p>This equation describes how the population grows over time if there are no restrictions on growth.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Part (b): Population after 4 days<\/h3>\n\n\n\n<p>We are given:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Initial population N0=1000N_0 = 1000N0\u200b=1000,<\/li>\n\n\n\n<li>After 1 day, the population is 1500.<\/li>\n<\/ul>\n\n\n\n<p>We need to find the value of kkk and use that to determine the population after 4 days.<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Step 1: Find the growth rate constant kkk:<\/strong><\/li>\n<\/ol>\n\n\n\n<p>Using the information that the population is 1500 after 1 day, we substitute into the formula:1500=1000ek\u22c511500 = 1000 e^{k \\cdot 1}1500=1000ek\u22c51<\/p>\n\n\n\n<p>Solving for kkk:15001000=ek\\frac{1500}{1000} = e^k10001500\u200b=ek1.5=ek1.5 = e^k1.5=ek<\/p>\n\n\n\n<p>Taking the natural logarithm of both sides:ln\u2061(1.5)=k\\ln(1.5) = kln(1.5)=kk\u22480.4055k \\approx 0.4055k\u22480.4055<\/p>\n\n\n\n<ol start=\"2\" class=\"wp-block-list\">\n<li><strong>Step 2: Find the population after 4 days:<\/strong><\/li>\n<\/ol>\n\n\n\n<p>Now that we know k\u22480.4055k \\approx 0.4055k\u22480.4055, we can use the formula to find the population after 4 days:N(4)=1000e0.4055\u22c54N(4) = 1000 e^{0.4055 \\cdot 4}N(4)=1000e0.4055\u22c54N(4)=1000e1.6222N(4) = 1000 e^{1.6222}N(4)=1000e1.6222N(4)\u22481000\u00d75.080N(4) \\approx 1000 \\times 5.080N(4)\u22481000\u00d75.080N(4)\u22485080&nbsp;mosquitoesN(4) \\approx 5080 \\text{ mosquitoes}N(4)\u22485080&nbsp;mosquitoes<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Part (c): Time until the population reaches 40,000 mosquitoes<\/h3>\n\n\n\n<p>We are asked to find the time ttt when the population reaches 40,000. Using the same formula:40,000=1000e0.4055\u22c5t40,000 = 1000 e^{0.4055 \\cdot t}40,000=1000e0.4055\u22c5t<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Step 1: Solve for ttt:<\/strong><\/li>\n<\/ol>\n\n\n\n<p>40,0001000=e0.4055\u22c5t\\frac{40,000}{1000} = e^{0.4055 \\cdot t}100040,000\u200b=e0.4055\u22c5t40=e0.4055\u22c5t40 = e^{0.4055 \\cdot t}40=e0.4055\u22c5t<\/p>\n\n\n\n<p>Taking the natural logarithm of both sides:ln\u2061(40)=0.4055\u22c5t\\ln(40) = 0.4055 \\cdot tln(40)=0.4055\u22c5tt=ln\u2061(40)0.4055t = \\frac{\\ln(40)}{0.4055}t=0.4055ln(40)\u200bt\u22483.68890.4055t \\approx \\frac{3.6889}{0.4055}t\u22480.40553.6889\u200bt\u22489.1&nbsp;dayst \\approx 9.1 \\text{ days}t\u22489.1&nbsp;days<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Final Answers:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>(a) The population function is N(t)=N0ektN(t) = N_0 e^{kt}N(t)=N0\u200bekt.<\/li>\n\n\n\n<li>(b) The population after 4 days is approximately <strong>5080 mosquitoes<\/strong>.<\/li>\n\n\n\n<li>(c) It will take approximately <strong>9.1 days<\/strong> for the population to reach 40,000 mosquitoes.<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner5-333.jpeg\" alt=\"\" class=\"wp-image-243466\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>The population of a colony of mosquitoes obeys the law of uninhibited growth: Use this information to answer parts (a) through (c): (a) If N is the population of the colony and t is the time in days, express N as a function of t. Consider No is the original amount at t = 0 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-243465","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/243465","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=243465"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/243465\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=243465"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=243465"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=243465"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}