{"id":212130,"date":"2025-05-06T06:16:51","date_gmt":"2025-05-06T06:16:51","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=212130"},"modified":"2025-05-06T07:12:02","modified_gmt":"2025-05-06T07:12:02","slug":"a-car-is-being-pulled-out-of-the-mud-by-two-forces-that-are-applied-by-the-two-ropes-shown-in-the-drawing","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/06\/a-car-is-being-pulled-out-of-the-mud-by-two-forces-that-are-applied-by-the-two-ropes-shown-in-the-drawing\/","title":{"rendered":"A car is being pulled out of the mud by two forces that are applied by the two ropes shown in the drawing"},"content":{"rendered":"\n<p>physics<br>A car is being pulled out of the mud by two forces that are applied by the two ropes shown in the drawing. The dashed line in the drawing bisects the 30.0\u00b0 angle. The magnitude of the force applied by each rope is 2900 newtons (N). (a) How much force would a single rope \u2026<br>Monday, January 27, 2014 at 8:38pm<\/p>\n\n\n\n<p>physics<br>Three deer, A, B, and C, are grazing in a field. Deer B is located 63.4 m from deer A at an angle of 50.8 \u00b0 north of west. Deer C is located 76.7 \u00b0 north of east relative to deer A. The distance between deer B and C is 96.7 m. What is the distance between deer A and C?<br>Monday, January 27, 2014 at 8:36pm<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">The correct answer and explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>Let&#8217;s solve the second problem:<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p><strong>Physics Problem (Deer A, B, and C):<\/strong><br>Three deer, A, B, and C, are grazing in a field.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Deer B is located <strong>63.4 m from A<\/strong>, at an angle of <strong>50.8\u00b0 north of west<\/strong>.<\/li>\n\n\n\n<li>Deer C is located at an angle of <strong>76.7\u00b0 north of east<\/strong> relative to A.<\/li>\n\n\n\n<li>The distance between deer B and C is <strong>96.7 m<\/strong>.<\/li>\n<\/ul>\n\n\n\n<p><strong>Question:<\/strong> What is the distance between deer A and C?<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step-by-step Explanation (300+ words):<\/h3>\n\n\n\n<p>We are given a triangle formed by deer A, B, and C. We can solve this problem using <strong>the Law of Cosines<\/strong>, but first, we need to understand the angles and construct a proper triangle.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Understand the geometry<\/h3>\n\n\n\n<p>Let\u2019s place <strong>Deer A at the origin (0, 0)<\/strong>.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Deer B is at <strong>63.4 m<\/strong> at <strong>50.8\u00b0 north of west<\/strong> \u21d2 this means 180\u00b0 \u2212 50.8\u00b0 = <strong>129.2\u00b0<\/strong> from the east (standard counterclockwise from east).<\/li>\n\n\n\n<li>Deer C is at some distance (unknown) at <strong>76.7\u00b0 north of east<\/strong>, which is simply <strong>76.7\u00b0<\/strong> from the east.<\/li>\n<\/ul>\n\n\n\n<p>Thus, we can calculate the angle \u2220BAC between vectors <strong>A\u2192B<\/strong> and <strong>A\u2192C<\/strong> using: Angle&nbsp;at&nbsp;A=76.7\u00b0+50.8\u00b0=127.5\u00b0\\text{Angle at A} = 76.7\u00b0 + 50.8\u00b0 = 127.5\u00b0<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Apply the Law of Cosines<\/h3>\n\n\n\n<p>Let:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>AB=63.4\u2009mAB = 63.4 \\, \\text{m}<\/li>\n\n\n\n<li>BC=96.7\u2009mBC = 96.7 \\, \\text{m}<\/li>\n\n\n\n<li>\u2220BAC = 127.5\u00b0<\/li>\n\n\n\n<li>We want to find AC=xAC = x<\/li>\n<\/ul>\n\n\n\n<p>From the Law of Cosines: BC2=AB2+AC2\u22122(AB)(AC)cos\u2061(\u2220BAC)BC^2 = AB^2 + AC^2 &#8211; 2(AB)(AC)\\cos(\\angle BAC)<\/p>\n\n\n\n<p>Substitute: 96.72=63.42+x2\u22122(63.4)(x)cos\u2061(127.5\u00b0)96.7^2 = 63.4^2 + x^2 &#8211; 2(63.4)(x)\\cos(127.5\u00b0)<\/p>\n\n\n\n<p>Evaluate the cosine: cos\u2061(127.5\u00b0)\u2248\u22120.6018\\cos(127.5\u00b0) \u2248 -0.6018<\/p>\n\n\n\n<p>Now plug in: 9355.69=4012.36+x2\u22122(63.4)(x)(\u22120.6018)9355.69 = 4012.36 + x^2 &#8211; 2(63.4)(x)(-0.6018) 9355.69=4012.36+x2+76.278&#215;9355.69 = 4012.36 + x^2 + 76.278x<\/p>\n\n\n\n<p>Rewriting: x2+76.278x+4012.36\u22129355.69=0x^2 + 76.278x + 4012.36 &#8211; 9355.69 = 0 x2+76.278x\u22125343.33=0x^2 + 76.278x &#8211; 5343.33 = 0<\/p>\n\n\n\n<p>Solve using the quadratic formula: x=\u221276.278\u00b176.2782+4(5343.33)2x = \\frac{-76.278 \\pm \\sqrt{76.278^2 + 4(5343.33)}}{2} x=\u221276.278\u00b15828.33+21373.322x = \\frac{-76.278 \\pm \\sqrt{5828.33 + 21373.32}}{2} x=\u221276.278\u00b127201.652x = \\frac{-76.278 \\pm \\sqrt{27201.65}}{2} x=\u221276.278\u00b1164.932x = \\frac{-76.278 \\pm 164.93}{2}<\/p>\n\n\n\n<p>Taking the positive root: x=88.652=44.33\u2009mx = \\frac{88.65}{2} = 44.33 \\, \\text{m}<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\u2705 Final Answer:<\/h3>\n\n\n\n<p><strong>The distance between deer A and C is approximately 44.3 me<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"<p>physicsA car is being pulled out of the mud by two forces that are applied by the two ropes shown in the drawing. The dashed line in the drawing bisects the 30.0\u00b0 angle. The magnitude of the force applied by each rope is 2900 newtons (N). (a) How much force would a single rope \u2026Monday, [&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-212130","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/212130","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=212130"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/212130\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=212130"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=212130"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=212130"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}