{"id":33503,"date":"2025-06-23T09:10:37","date_gmt":"2025-06-23T09:10:37","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=33503"},"modified":"2025-06-23T09:10:39","modified_gmt":"2025-06-23T09:10:39","slug":"assume-that-the-half-life-of-pennies-is-1-shake","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/assume-that-the-half-life-of-pennies-is-1-shake\/","title":{"rendered":"Assume that the half-life of pennies is 1 shake."},"content":{"rendered":"\n<p>Assume that the half-life of pennies is 1 shake. Show your work answering the following questions: If you start with 600 pennies, how many would be left unchanged after 4 shakes? You come across an unfinished lab that has 6 pennies with heads up and 94 with tails up. How many shakes did the unfinished group do?<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Part 1: How many pennies remain unchanged after 4 shakes?<\/h3>\n\n\n\n<p><strong>Given:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Half-life = 1 shake<\/li>\n\n\n\n<li>Starting pennies = 600<\/li>\n<\/ul>\n\n\n\n<p>The concept of half-life means that after each shake, half of the remaining pennies stay unchanged (heads up). We can use the formula:N=N0\u00d7(12)tN = N_0 \\times \\left( \\frac{1}{2} \\right)^tN=N0\u200b\u00d7(21\u200b)t<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>NNN = number of unchanged pennies after ttt shakes<\/li>\n\n\n\n<li>N0N_0N0\u200b = initial number of pennies = 600<\/li>\n\n\n\n<li>ttt = number of shakes = 4<\/li>\n<\/ul>\n\n\n\n<p><strong>Calculation:<\/strong>N=600\u00d7(12)4=600\u00d7116=37.5N = 600 \\times \\left( \\frac{1}{2} \\right)^4 = 600 \\times \\frac{1}{16} = 37.5N=600\u00d7(21\u200b)4=600\u00d7161\u200b=37.5<\/p>\n\n\n\n<p>Since we cannot have half a penny, we approximate to the nearest whole number. So, <strong>38 pennies remain unchanged after 4 shakes<\/strong>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Part 2: How many shakes did the unfinished group do?<\/h3>\n\n\n\n<p><strong>Given:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Heads up (unchanged) = 6 pennies<\/li>\n\n\n\n<li>Total pennies = 6 heads + 94 tails = 100 pennies<\/li>\n\n\n\n<li>Half-life = 1 shake<\/li>\n<\/ul>\n\n\n\n<p>We use the same formula:N=N0\u00d7(12)tN = N_0 \\times \\left( \\frac{1}{2} \\right)^tN=N0\u200b\u00d7(21\u200b)t<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>NNN = 6 unchanged pennies<\/li>\n\n\n\n<li>N0N_0N0\u200b = 100 pennies<\/li>\n\n\n\n<li>ttt = unknown number of shakes<\/li>\n<\/ul>\n\n\n\n<p><strong>Solving for ttt:<\/strong>6=100\u00d7(12)t6 = 100 \\times \\left( \\frac{1}{2} \\right)^t6=100\u00d7(21\u200b)t<\/p>\n\n\n\n<p>Divide both sides by 100:0.06=(12)t0.06 = \\left( \\frac{1}{2} \\right)^t0.06=(21\u200b)t<\/p>\n\n\n\n<p>Take logarithms of both sides:log\u2061(0.06)=t\u00d7log\u2061(0.5)\\log(0.06) = t \\times \\log(0.5)log(0.06)=t\u00d7log(0.5)<\/p>\n\n\n\n<p>Solve for ttt:t=log\u2061(0.06)log\u2061(0.5)\u2248\u22121.2218\u22120.3010\u22484.06t = \\frac{\\log(0.06)}{\\log(0.5)} \\approx \\frac{-1.2218}{-0.3010} \\approx 4.06t=log(0.5)log(0.06)\u200b\u2248\u22120.3010\u22121.2218\u200b\u22484.06<\/p>\n\n\n\n<p>So, approximately <strong>4 shakes<\/strong> were completed.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Conclusion:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>After 4 shakes, 38 pennies remain unchanged out of 600.<\/li>\n\n\n\n<li>The unfinished group completed approximately 4 shakes, based on 6 out of 100 pennies remaining unchanged.<\/li>\n<\/ul>\n\n\n\n<p>This follows the principle of exponential decay where each shake reduces the unchanged group by half.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"852\" height=\"1024\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-518.jpeg\" alt=\"\" class=\"wp-image-33504\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-518.jpeg 852w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-518-250x300.jpeg 250w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-518-768x923.jpeg 768w\" sizes=\"auto, (max-width: 852px) 100vw, 852px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Assume that the half-life of pennies is 1 shake. Show your work answering the following questions: If you start with 600 pennies, how many would be left unchanged after 4 shakes? You come across an unfinished lab that has 6 pennies with heads up and 94 with tails up. How many shakes did the unfinished [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-33503","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/33503","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=33503"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/33503\/revisions"}],"predecessor-version":[{"id":33505,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/33503\/revisions\/33505"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=33503"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=33503"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=33503"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}