The thickness of a US penny is 1.52 mm = 0.00152 m The distance from the surface of the earth to the moon is 384402 km = 384402000 m Each day you have (in total) double the number of pennies you have from the previous day, starting with one penny. That is: Today (day 0) you have 1 penny. Tomorrow (day 1) you have 2 pennies. Day 2 you have 4 pennies. Day 3 you have 8 pennies. If you stack the pennies together vertically, on what day will the pennies reach the moon?<\/p>\n\n\n\n
The correct answer and explanation is:<\/strong><\/mark><\/p>\n\n\n\n To solve this problem, we need to determine how many doublings<\/strong> of a single penny (1.52 mm thick) are required to create a stack that reaches from the Earth to the Moon (384,402,000 meters).<\/p>\n\n\n\n You start with 1 penny<\/strong> and double<\/strong> the number each day:<\/p>\n\n\n\n Each penny is 0.00152 meters<\/strong> thick.<\/p>\n\n\n\n So, on day n<\/em>, the stack height is: Heightn=2n\u00d70.00152 meters\\text{Height}_n = 2^n \\times 0.00152 \\text{ meters}<\/p>\n\n\n\n We want this to be at least 384,402,000 meters<\/strong> (distance to the Moon).<\/p>\n\n\n\n So solve: 2n\u00d70.00152\u2265384,402,0002^n \\times 0.00152 \\geq 384,402,000<\/p>\n\n\n\n Divide both sides by 0.00152: 2n\u2265384,402,0000.00152=252,894,736,842.12^n \\geq \\frac{384,402,000}{0.00152} = 252,894,736,842.1<\/p>\n\n\n\n Now take the base-2 logarithm of both sides: n\u2265log\u20612(252,894,736,842.1)\u224837.9n \\geq \\log_2(252,894,736,842.1) \\approx 37.9<\/p>\n\n\n\n Since n<\/em> must be an integer<\/strong>, we round up: n=38n = 38<\/p>\n\n\n\n This is a classic exponential growth problem that illustrates how quickly numbers grow when they double each time. Starting with a single U.S. penny (1.52 mm thick), each day you double the total number of pennies you have. On day 0, there is 1 penny, day 1 has 2, day 2 has 4, and so on, meaning the number of pennies on day n<\/em> is 2n2^n. If you were to stack all the pennies you own vertically, you would multiply the number of pennies by the thickness of one to get the total height.<\/p>\n\n\n\n The goal is to find when that stack reaches the Moon \u2014 about 384,402 kilometers or 384,402,000 meters away. Since one penny is 0.00152 meters thick, the stack height on day n<\/em> is 2n\u00d70.001522^n \\times 0.00152 meters. Setting this equal to the distance to the Moon and solving for n<\/em> yields approximately 37.9. Since you can\u2019t have a fraction of a day in this context, we round up to 38.<\/p>\n\n\n\n So, on day 38<\/strong>, the vertical stack of pennies will exceed the distance from Earth to the Moon. This result is a powerful example of exponential growth: in just 38 doublings \u2014 a little over a month \u2014 a stack starting from a single coin can reach space!<\/p>\n","protected":false},"excerpt":{"rendered":" The thickness of a US penny is 1.52 mm = 0.00152 m The distance from the surface of the earth to the moon is 384402 km = 384402000 m Each day you have (in total) double the number of pennies you have from the previous day, starting with one penny. That is: Today (day 0) […]<\/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-22423","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/22423","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=22423"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/22423\/revisions"}],"predecessor-version":[{"id":22424,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/22423\/revisions\/22424"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=22423"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=22423"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=22423"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n\n\n\nStep 1: Understand what doubles daily<\/strong><\/h3>\n\n\n\n
\n
\n\n\n\nStep 2: Stack height formula<\/strong><\/h3>\n\n\n\n
\n\n\n\nStep 3: Round up<\/strong><\/h3>\n\n\n\n
\n\n\n\n\u2705 Answer: Day 38<\/strong><\/h3>\n\n\n\n
\n\n\n\nExplanation (300 words)<\/strong><\/h3>\n\n\n\n