{"id":241727,"date":"2025-07-03T20:26:58","date_gmt":"2025-07-03T20:26:58","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=241727"},"modified":"2025-07-03T20:27:00","modified_gmt":"2025-07-03T20:27:00","slug":"a-plane-is-located-at-c-on-the-diagram","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/03\/a-plane-is-located-at-c-on-the-diagram\/","title":{"rendered":"A plane is located at C on the diagram"},"content":{"rendered":"\n<p>A plane is located at C on the diagram. There are two towers located at A and B. The distance between the towers is 7,600 feet, and the angles of elevation are given. A 16\u00b0 B 24\u00b0 Tower 1 7600 ft Tower 2 D drawing not to scale C a. Find BC, the distance from Tower 2 to the plane, to the nearest foot. b. Find CD, the height of the plane from the ground, to the nearest foot.<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To solve this problem, we will use trigonometry, specifically the tangent function, which relates angles of elevation to distances.<\/p>\n\n\n\n<p><strong>Given:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Distance between the two towers AB=7600AB = 7600AB=7600 feet.<\/li>\n\n\n\n<li>Angle of elevation at tower A \u2220A=16\u2218\\angle A = 16^\\circ\u2220A=16\u2218.<\/li>\n\n\n\n<li>Angle of elevation at tower B \u2220B=24\u2218\\angle B = 24^\\circ\u2220B=24\u2218.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Part a: Find BC (distance from Tower 2 to the plane)<\/h3>\n\n\n\n<p>We can use the law of sines or break the problem into smaller right triangles. Let\u2019s start by considering the triangles formed by the towers and the plane.<\/p>\n\n\n\n<p>From the diagram:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Height\u00a0at\u00a0tower\u00a0A=hA\\text{Height at tower A} = h_AHeight\u00a0at\u00a0tower\u00a0A=hA\u200b<\/li>\n\n\n\n<li>Height\u00a0at\u00a0tower\u00a0B=hB\\text{Height at tower B} = h_BHeight\u00a0at\u00a0tower\u00a0B=hB\u200b<\/li>\n\n\n\n<li>BC=xBC = xBC=x<\/li>\n<\/ul>\n\n\n\n<p>The tangent of the angles gives the relationship between the heights and horizontal distances for each tower:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>For tower A: tan\u2061(16\u2218)=hAx\\tan(16^\\circ) = \\frac{h_A}{x}tan(16\u2218)=xhA\u200b\u200b Therefore, hA=x\u22c5tan\u2061(16\u2218)h_A = x \\cdot \\tan(16^\\circ)hA\u200b=x\u22c5tan(16\u2218)<\/li>\n\n\n\n<li>For tower B: tan\u2061(24\u2218)=hB7600\u2212x\\tan(24^\\circ) = \\frac{h_B}{7600 &#8211; x}tan(24\u2218)=7600\u2212xhB\u200b\u200b Therefore, hB=(7600\u2212x)\u22c5tan\u2061(24\u2218)h_B = (7600 &#8211; x) \\cdot \\tan(24^\\circ)hB\u200b=(7600\u2212x)\u22c5tan(24\u2218)<\/li>\n<\/ol>\n\n\n\n<p>Now, the difference between the heights at A and B is the height of the plane hCh_ChC\u200b:hC=hA\u2212hBh_C = h_A &#8211; h_BhC\u200b=hA\u200b\u2212hB\u200b<\/p>\n\n\n\n<p>Substitute the expressions for hAh_AhA\u200b and hBh_BhB\u200b:hC=x\u22c5tan\u2061(16\u2218)\u2212(7600\u2212x)\u22c5tan\u2061(24\u2218)h_C = x \\cdot \\tan(16^\\circ) &#8211; (7600 &#8211; x) \\cdot \\tan(24^\\circ)hC\u200b=x\u22c5tan(16\u2218)\u2212(7600\u2212x)\u22c5tan(24\u2218)<\/p>\n\n\n\n<p>Solve for xxx using known values for the tangents of the angles and the fact that hC=0h_C = 0hC\u200b=0 (since the plane is at the same height at both towers):x\u22c5tan\u2061(16\u2218)=(7600\u2212x)\u22c5tan\u2061(24\u2218)x \\cdot \\tan(16^\\circ) = (7600 &#8211; x) \\cdot \\tan(24^\\circ)x\u22c5tan(16\u2218)=(7600\u2212x)\u22c5tan(24\u2218)<\/p>\n\n\n\n<p>Plug in values for the tangents:x\u22c50.2867=(7600\u2212x)\u22c50.4452x \\cdot 0.2867 = (7600 &#8211; x) \\cdot 0.4452x\u22c50.2867=(7600\u2212x)\u22c50.4452<\/p>\n\n\n\n<p>Simplifying and solving for xxx:0.2867x=7600\u22c50.4452\u2212x\u22c50.44520.2867x = 7600 \\cdot 0.4452 &#8211; x \\cdot 0.44520.2867x=7600\u22c50.4452\u2212x\u22c50.44520.2867x+0.4452x=7600\u22c50.44520.2867x + 0.4452x = 7600 \\cdot 0.44520.2867x+0.4452x=7600\u22c50.44520.7319x=3381.720.7319x = 3381.720.7319x=3381.72x\u22483381.720.7319\u22484620&nbsp;feetx \\approx \\frac{3381.72}{0.7319} \\approx 4620 \\text{ feet}x\u22480.73193381.72\u200b\u22484620&nbsp;feet<\/p>\n\n\n\n<p>So, the distance from Tower 2 to the plane, BCBCBC, is approximately <strong>4,620 feet<\/strong>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Part b: Find CD (height of the plane from the ground)<\/h3>\n\n\n\n<p>To find the height of the plane, we can substitute the value of xxx into one of the previous equations. Using the equation for hAh_AhA\u200b:hA=x\u22c5tan\u2061(16\u2218)h_A = x \\cdot \\tan(16^\\circ)hA\u200b=x\u22c5tan(16\u2218)<\/p>\n\n\n\n<p>Substitute x=4620x = 4620x=4620:hA=4620\u22c50.2867\u22481322&nbsp;feeth_A = 4620 \\cdot 0.2867 \\approx 1322 \\text{ feet}hA\u200b=4620\u22c50.2867\u22481322&nbsp;feet<\/p>\n\n\n\n<p>So, the height of the plane from the ground, CDCDCD, is approximately <strong>1,322 feet<\/strong>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Conclusion:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>BC<\/strong> (distance from Tower 2 to the plane) is approximately <strong>4,620 feet<\/strong>.<\/li>\n\n\n\n<li><strong>CD<\/strong> (height of the plane from the ground) is approximately <strong>1,322 feet<\/strong>.<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner5-173.jpeg\" alt=\"\" class=\"wp-image-241728\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>A plane is located at C on the diagram. There are two towers located at A and B. The distance between the towers is 7,600 feet, and the angles of elevation are given. A 16\u00b0 B 24\u00b0 Tower 1 7600 ft Tower 2 D drawing not to scale C a. Find BC, the distance from [&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-241727","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/241727","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=241727"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/241727\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=241727"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=241727"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=241727"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}