{"id":235768,"date":"2025-06-15T18:31:11","date_gmt":"2025-06-15T18:31:11","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=235768"},"modified":"2025-06-15T18:31:13","modified_gmt":"2025-06-15T18:31:13","slug":"a-man-starts-walking-north-at-3-ft-s-from-point-p","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/15\/a-man-starts-walking-north-at-3-ft-s-from-point-p\/","title":{"rendered":"A man starts walking north at 3 ft\/s from point P."},"content":{"rendered":"\n

A man starts walking north at 3 ft\/s from point P. Five minutes later, a woman starts walking south at 4 ft\/s from a point 500 ft due east of P. At what rate are the people moving apart 15 minutes after the woman starts walking? (Round your answer to two decimal places.) ft\/s At noon, ship A is 70 km west of ship B. Ship A is sailing south at 35 km\/h and ship B is sailing north at 25 km\/h. How fast is the distance between the ships changing at 4:00 PM? (Round your answer to one decimal place.) km\/h<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

Problem 1: Man and Woman Walking Apart<\/h3>\n\n\n\n

Let the coordinate system place point P at the origin (0,0)(0, 0)(0,0).
The man walks north at 333 ft\/s from point P, so his position at time ttt seconds after the woman starts walking is:(x1,y1)=(0,3(t+300))(x_1, y_1) = (0, 3(t + 300))(x1\u200b,y1\u200b)=(0,3(t+300))<\/p>\n\n\n\n

(The man starts 5 minutes = 300 seconds earlier.)<\/p>\n\n\n\n

The woman starts 500 ft east of P and walks south at 444 ft\/s:(x2,y2)=(500,\u22124t)(x_2, y_2) = (500, -4t)(x2\u200b,y2\u200b)=(500,\u22124t)<\/p>\n\n\n\n

The distance DDD between the two people is:D=(x2\u2212x1)2+(y2\u2212y1)2=(500)2+(3t+300+4t)2=250000+(7t+300)2D = \\sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2} = \\sqrt{(500)^2 + (3t + 300 + 4t)^2} = \\sqrt{250000 + (7t + 300)^2}D=(x2\u200b\u2212x1\u200b)2+(y2\u200b\u2212y1\u200b)2\u200b=(500)2+(3t+300+4t)2\u200b=250000+(7t+300)2\u200b<\/p>\n\n\n\n

Differentiate with respect to ttt:dDdt=12\u22c5(250000+(7t+300)2)\u22121\/2\u22c52(7t+300)\u22c57=7(7t+300)250000+(7t+300)2\\frac{dD}{dt} = \\frac{1}{2} \\cdot \\left(250000 + (7t + 300)^2\\right)^{-1\/2} \\cdot 2(7t + 300) \\cdot 7 = \\frac{7(7t + 300)}{\\sqrt{250000 + (7t + 300)^2}}dtdD\u200b=21\u200b\u22c5(250000+(7t+300)2)\u22121\/2\u22c52(7t+300)\u22c57=250000+(7t+300)2\u200b7(7t+300)\u200b<\/p>\n\n\n\n

At t=900t = 900t=900 seconds (15 minutes):dDdt=7(7\u22c5900+300)250000+(7\u22c5900+300)2=7(6300+300)250000+66002=7\u22c56600250000+43560000=4620043810000\u2248462006617.41\u22486.98 ft\/s\\frac{dD}{dt} = \\frac{7(7 \\cdot 900 + 300)}{\\sqrt{250000 + (7 \\cdot 900 + 300)^2}} = \\frac{7(6300 + 300)}{\\sqrt{250000 + 6600^2}} = \\frac{7 \\cdot 6600}{\\sqrt{250000 + 43560000}} = \\frac{46200}{\\sqrt{43810000}} \u2248 \\frac{46200}{6617.41} \u2248 6.98 \\text{ ft\/s}dtdD\u200b=250000+(7\u22c5900+300)2\u200b7(7\u22c5900+300)\u200b=250000+66002\u200b7(6300+300)\u200b=250000+43560000\u200b7\u22c56600\u200b=43810000\u200b46200\u200b\u22486617.4146200\u200b\u22486.98 ft\/s<\/p>\n\n\n\n

Answer: 6.98 ft\/s<\/strong><\/p>\n\n\n\n

\n\n\n\n

Problem 2: Two Ships<\/h3>\n\n\n\n

At noon:<\/p>\n\n\n\n

    \n
  • Ship A is 70 km west of Ship B.<\/li>\n\n\n\n
  • Let Ship B be at origin (0, 0).<\/li>\n\n\n\n
  • Ship A starts at (-70, 0).<\/li>\n<\/ul>\n\n\n\n

    Ship A sails south at 35 km\/h
    Ship B sails north at 25 km\/h<\/p>\n\n\n\n

    Let t=4t = 4t=4 hours after noon.
    Position of Ship A: (\u221270,\u221235t)=(\u221270,\u2212140)(-70, -35t) = (-70, -140)(\u221270,\u221235t)=(\u221270,\u2212140)
    Position of Ship B: (0,25t)=(0,100)(0, 25t) = (0, 100)(0,25t)=(0,100)<\/p>\n\n\n\n

    Distance between ships:D=(\u221270)2+(\u2212140\u2212100)2=4900+57600=62500=250 kmD = \\sqrt{(-70)^2 + (-140 – 100)^2} = \\sqrt{4900 + 57600} = \\sqrt{62500} = 250 \\text{ km}D=(\u221270)2+(\u2212140\u2212100)2\u200b=4900+57600\u200b=62500\u200b=250 km<\/p>\n\n\n\n

    Differentiate:D2=x2+y2D^2 = x^2 + y^2D2=x2+y22D\u22c5dDdt=2x\u22c5dxdt+2y\u22c5dydt2D \\cdot \\frac{dD}{dt} = 2x \\cdot \\frac{dx}{dt} + 2y \\cdot \\frac{dy}{dt}2D\u22c5dtdD\u200b=2x\u22c5dtdx\u200b+2y\u22c5dtdy\u200b<\/p>\n\n\n\n

    At t=4t = 4t=4:<\/p>\n\n\n\n

      \n
    • x=\u221270x = -70x=\u221270, dx\/dt=0dx\/dt = 0dx\/dt=0<\/li>\n\n\n\n
    • y=\u2212140\u2212100=\u2212240y = -140 – 100 = -240y=\u2212140\u2212100=\u2212240<\/li>\n\n\n\n
    • dy\/dt=\u221235\u221225=\u221260dy\/dt = -35 – 25 = -60dy\/dt=\u221235\u221225=\u221260 km\/h<\/li>\n<\/ul>\n\n\n\n

      Substitute:2(250)\u22c5dDdt=2(\u2212240)(\u221260)\u21d2500\u22c5dDdt=28800\u21d2dDdt=28800500=57.6 km\/h2(250) \\cdot \\frac{dD}{dt} = 2(-240)(-60) \\Rightarrow 500 \\cdot \\frac{dD}{dt} = 28800 \\Rightarrow \\frac{dD}{dt} = \\frac{28800}{500} = 57.6 \\text{ km\/h}2(250)\u22c5dtdD\u200b=2(\u2212240)(\u221260)\u21d2500\u22c5dtdD\u200b=28800\u21d2dtdD\u200b=50028800\u200b=57.6 km\/h<\/p>\n\n\n\n

      Answer: 57.6 km\/h<\/strong><\/p>\n\n\n\n

      \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

      A man starts walking north at 3 ft\/s from point P. Five minutes later, a woman starts walking south at 4 ft\/s from a point 500 ft due east of P. At what rate are the people moving apart 15 minutes after the woman starts walking? (Round your answer to two decimal places.) ft\/s At […]<\/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-235768","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/235768","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=235768"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/235768\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=235768"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=235768"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=235768"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}