Mikayla is training for a half-marathon. At the beginning of her training, she is able to run 2 miles without stopping. Her goal is to increase her distance by three quarters of a mile each week.<\/p>\n\n\n\n
a) Write an equation to represent the number of miles Mikayla is able to run after each week. Define your variables.\n\n b) If a half-marathon (13.1 miles) is 12 weeks after the start of Mikayla's training program, will she be ready?<\/code><\/pre>\n\n\n\nThe Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\nLet’s solve this in two parts:<\/p>\n\n\n\n
\n\n\n\na) Write an equation to represent the number of miles Mikayla is able to run after each week. Define your variables.<\/strong><\/h3>\n\n\n\nLet:<\/p>\n\n\n\n
\n- ww = number of weeks since Mikayla started training<\/li>\n\n\n\n
- M(w)M(w) = number of miles Mikayla can run after ww weeks<\/li>\n<\/ul>\n\n\n\n
She starts at 2 miles and increases her distance by three-quarters of a mile (0.75 miles)<\/strong> each week. This is an arithmetic sequence<\/strong>, and the equation to model this is: M(w)=2+0.75wM(w) = 2 + 0.75w<\/p>\n\n\n\n
\n\n\n\nb) If a half-marathon (13.1 miles) is 12 weeks after the start of Mikayla’s training program, will she be ready?<\/strong><\/h3>\n\n\n\nSubstitute w=12w = 12 into the equation: M(12)=2+0.75(12)=2+9=11 milesM(12) = 2 + 0.75(12) = 2 + 9 = 11 \\text{ miles}<\/p>\n\n\n\n
So, after 12 weeks, Mikayla will be able to run 11 miles<\/strong>.<\/p>\n\n\n\nA half-marathon is 13.1 miles<\/strong>, which means: 13.1\u221211=2.1 miles short13.1 – 11 = 2.1 \\text{ miles short}<\/p>\n\n\n\n
\n\n\n\nExplanation (300+ words):<\/strong><\/h3>\n\n\n\nTo determine Mikayla\u2019s readiness for a half-marathon, we start by modeling her weekly progress. Mikayla begins by running 2 miles and adds three-quarters of a mile every week. This constant weekly increase indicates that her training follows an arithmetic pattern. The general form of an arithmetic sequence is: M(w)=initial value+(rate of change)\u00d7wM(w) = \\text{initial value} + (\\text{rate of change}) \\times w<\/p>\n\n\n\n
For Mikayla, the initial value is 2 miles, and the rate of change is 0.75 miles per week. So, her running capability each week is modeled by: M(w)=2+0.75wM(w) = 2 + 0.75w<\/p>\n\n\n\n
Now, the question is whether she can run 13.1 miles (the distance of a half-marathon) by week 12. Plugging in w=12w = 12 into the equation: M(12)=2+0.75(12)=2+9=11 milesM(12) = 2 + 0.75(12) = 2 + 9 = 11 \\text{ miles}<\/p>\n\n\n\n
This means that after 12 weeks of training, Mikayla will be able to run 11 miles continuously. However, since a half-marathon is 13.1 miles, she falls short by 2.1 miles. While 11 miles is a strong achievement, it\u2019s not quite the full race distance.<\/p>\n\n\n\n
In conclusion, Mikayla will not be fully ready<\/strong> to run a half-marathon after 12 weeks at her current rate of increase. To be fully prepared, she would need more time\u2014specifically, we can solve: 13.1=2+0.75w\u21d20.75w=11.1\u21d2w=11.10.75=14.8 weeks13.1 = 2 + 0.75w \\Rightarrow 0.75w = 11.1 \\Rightarrow w = \\frac{11.1}{0.75} = 14.8 \\text{ weeks}<\/p>\n\n\n\nSo she would need about 15 weeks<\/strong> total to be ready.<\/p>\n","protected":false},"excerpt":{"rendered":"Mikayla is training for a half-marathon. At the beginning of her training, she is able to run 2 miles without stopping. Her goal is to increase her distance by three quarters of a mile each week. The Correct Answer and Explanation is: Let’s solve this in two parts: a) Write an equation to represent the […]<\/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-216714","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/216714","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=216714"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/216714\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=216714"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=216714"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=216714"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}