{"id":182751,"date":"2025-01-14T19:20:51","date_gmt":"2025-01-14T19:20:51","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=182751"},"modified":"2025-01-14T19:20:53","modified_gmt":"2025-01-14T19:20:53","slug":"a-man-starts-walking-north-at-4-ft-s-from-a-point-p","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/01\/14\/a-man-starts-walking-north-at-4-ft-s-from-a-point-p\/","title":{"rendered":"A man starts walking north at 4 ft\/s from a point P"},"content":{"rendered":"\n<p>A man starts walking north at 4 ft\/s from a point P. Five minutes later a woman starts walking south at 5 ft\/s from a point 500 ft due east of . At what rate are the people moving apart 15 min after the woman starts walking?<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>Let&#8217;s break this problem into parts and solve it step by step using related rates.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Define variables<\/h3>\n\n\n\n<p>Let ( t ) represent the time in seconds since the woman started walking. The problem states that the woman starts walking 5 minutes after the man, which is equivalent to 300 seconds. So, the time ( t = 0 ) corresponds to 5 minutes after the man starts walking.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Let ( x(t) ) represent the east-west distance between the two people, which is constant because the man is walking north, and the woman is walking south. Initially, ( x = 500 ) ft, and this value does not change.<\/li>\n\n\n\n<li>Let ( y_m(t) ) represent the position of the man along the north-south axis (measured in feet). Since the man starts walking north at 4 ft\/s, his position at time ( t ) is given by ( y_m(t) = 4(t + 300) ), because the man has been walking for 300 seconds before the woman starts walking.<\/li>\n\n\n\n<li>Let ( y_w(t) ) represent the position of the woman along the north-south axis. Since she starts walking south at 5 ft\/s, her position at time ( t ) is given by ( y_w(t) = 5t ).<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Use the distance formula<\/h3>\n\n\n\n<p>The distance ( D(t) ) between the two people at time ( t ) is given by the Pythagorean theorem, because the two people are moving along perpendicular paths (north-south and east-west):<\/p>\n\n\n\n<p>[<br>D(t) = \\sqrt{x(t)^2 + (y_m(t) &#8211; y_w(t))^2}<br>]<\/p>\n\n\n\n<p>Since ( x(t) = 500 ) ft is constant, we have:<\/p>\n\n\n\n<p>[<br>D(t) = \\sqrt{500^2 + (y_m(t) &#8211; y_w(t))^2}<br>]<\/p>\n\n\n\n<p>Substitute the expressions for ( y_m(t) ) and ( y_w(t) ):<\/p>\n\n\n\n<p>[<br>D(t) = \\sqrt{500^2 + \\left( 4(t + 300) &#8211; 5t \\right)^2}<br>]<\/p>\n\n\n\n<p>Simplify the expression inside the parentheses:<\/p>\n\n\n\n<p>[<br>D(t) = \\sqrt{500^2 + \\left( 4t + 1200 &#8211; 5t \\right)^2}<br>]<br>[<br>D(t) = \\sqrt{500^2 + (-t + 1200)^2}<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: Differentiate with respect to time<\/h3>\n\n\n\n<p>To find the rate at which the distance between the two people is changing, differentiate ( D(t) ) with respect to time ( t ):<\/p>\n\n\n\n<p>[<br>\\frac{dD}{dt} = \\frac{d}{dt} \\left( \\sqrt{500^2 + (-t + 1200)^2} \\right)<br>]<\/p>\n\n\n\n<p>Apply the chain rule:<\/p>\n\n\n\n<p>[<br>\\frac{dD}{dt} = \\frac{1}{2} \\left( 500^2 + (-t + 1200)^2 \\right)^{-1\/2} \\cdot 2(-t + 1200)(-1)<br>]<br>[<br>\\frac{dD}{dt} = \\frac{(-t + 1200)}{\\sqrt{500^2 + (-t + 1200)^2}}<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 4: Calculate at ( t = 900 ) seconds<\/h3>\n\n\n\n<p>The problem asks for the rate at which the people are moving apart 15 minutes (900 seconds) after the woman starts walking. Plug ( t = 900 ) into the expression:<\/p>\n\n\n\n<p>[<br>\\frac{dD}{dt} = \\frac{(-900 + 1200)}{\\sqrt{500^2 + (-900 + 1200)^2}}<br>]<br>[<br>\\frac{dD}{dt} = \\frac{300}{\\sqrt{500^2 + 300^2}}<br>]<\/p>\n\n\n\n<p>Now calculate the square root:<\/p>\n\n\n\n<p>[<br>\\frac{dD}{dt} = \\frac{300}{\\sqrt{250000 + 90000}} = \\frac{300}{\\sqrt{340000}} \\approx \\frac{300}{583.095}<br>]<\/p>\n\n\n\n<p>[<br>\\frac{dD}{dt} \\approx 0.514 \\, \\text{ft\/s}<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Final Answer:<\/h3>\n\n\n\n<p>The rate at which the two people are moving apart 15 minutes after the woman starts walking is approximately <strong>0.514 ft\/s<\/strong>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A man starts walking north at 4 ft\/s from a point P. Five minutes later a woman starts walking south at 5 ft\/s from a point 500 ft due east of . At what rate are the people moving apart 15 min after the woman starts walking? The Correct Answer and Explanation is : Let&#8217;s [&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-182751","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/182751","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=182751"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/182751\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=182751"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=182751"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=182751"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}