{"id":207486,"date":"2025-04-24T19:01:39","date_gmt":"2025-04-24T19:01:39","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=207486"},"modified":"2025-04-24T19:01:41","modified_gmt":"2025-04-24T19:01:41","slug":"two-hikers-on-opposite-sides-of-a-canyon-each-stand-precisely-525-meters-above-the-canyon-floor","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/04\/24\/two-hikers-on-opposite-sides-of-a-canyon-each-stand-precisely-525-meters-above-the-canyon-floor\/","title":{"rendered":"Two hikers on opposite sides of a canyon each stand precisely 525 meters above the canyon floor"},"content":{"rendered":"\n

Two hikers on opposite sides of a canyon each stand precisely 525 meters above the canyon floor. They each sight a landmark on the canyon floor on a line directly between them. The angles of depression from each hiker to the landmark meter are 37\u00b0 and 21\u00b0. How far apart are the hikers? Round your answer to the nearest whole meter. 1064 m 2064 m 2063 m 2065 m<\/p>\n\n\n\n

The correct answer and explanation is :<\/strong><\/mark><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

To find how far apart the two hikers are, we can model this as a trigonometry problem involving right triangles.<\/p>\n\n\n\n

\n\n\n\n

Step 1: Visualize the problem<\/strong><\/h3>\n\n\n\n
    \n
  • Each hiker is on opposite sides of a canyon and 525 meters above<\/strong> the canyon floor.<\/li>\n\n\n\n
  • There\u2019s a landmark directly on the canyon floor<\/strong> between them.<\/li>\n\n\n\n
  • The angles of depression<\/strong> from each hiker to the landmark are 37\u00b0<\/strong> and 21\u00b0<\/strong>.<\/li>\n<\/ul>\n\n\n\n

    This gives us two right triangles<\/strong> sharing a base (the horizontal ground between the two hikers), with one side at 525 meters high and angles of depression of 37\u00b0 and 21\u00b0 respectively.<\/p>\n\n\n\n

    \n\n\n\n

    Step 2: Apply trigonometry<\/strong><\/h3>\n\n\n\n

    Using right triangle trigonometry<\/strong>, particularly the tangent<\/strong> function:<\/p>\n\n\n\n

    \n

    (\\tan(\\theta) = \\frac{\\text{opposite}}{\\text{adjacent}})<\/p>\n<\/blockquote>\n\n\n\n

    For each hiker:<\/p>\n\n\n\n

    Let:<\/p>\n\n\n\n

      \n
    • (x_1) = horizontal distance from the hiker at 37\u00b0 to the landmark<\/li>\n\n\n\n
    • (x_2) = horizontal distance from the hiker at 21\u00b0 to the landmark<\/li>\n<\/ul>\n\n\n\n

      Using the tangent definition:<\/p>\n\n\n\n

        \n
      • (\\tan(37^\\circ) = \\frac{525}{x_1} \\Rightarrow x_1 = \\frac{525}{\\tan(37^\\circ)})<\/li>\n\n\n\n
      • (\\tan(21^\\circ) = \\frac{525}{x_2} \\Rightarrow x_2 = \\frac{525}{\\tan(21^\\circ)})<\/li>\n<\/ul>\n\n\n\n
        \n\n\n\n

        Step 3: Compute distances<\/strong><\/h3>\n\n\n\n
          \n
        • (\\tan(37^\\circ) \\approx 0.7536 \\Rightarrow x_1 \\approx \\frac{525}{0.7536} \\approx 696.97 \\, \\text{m})<\/li>\n\n\n\n
        • (\\tan(21^\\circ) \\approx 0.3839 \\Rightarrow x_2 \\approx \\frac{525}{0.3839} \\approx 1367.68 \\, \\text{m})<\/li>\n<\/ul>\n\n\n\n

          Add them to find the total distance between hikers:<\/p>\n\n\n\n

          \n

          (x_1 + x_2 \\approx 696.97 + 1367.68 \\approx 2064.65)<\/p>\n<\/blockquote>\n\n\n\n

          \n\n\n\n

          Step 4: Final Answer<\/strong><\/h3>\n\n\n\n

          Rounded to the nearest whole meter, the distance between the hikers is 2065 meters<\/strong>.<\/p>\n\n\n\n

          \n\n\n\n

          \u2705 Correct Answer: 2065 m<\/strong> \u2705<\/h3>\n\n\n\n

          This trigonometric method works because it relies on basic right triangle relationships and the consistent elevation of both hikers. It\u2019s a classic example of applying tangent in real-world navigation or surveying contexts.<\/p>\n","protected":false},"excerpt":{"rendered":"

          Two hikers on opposite sides of a canyon each stand precisely 525 meters above the canyon floor. They each sight a landmark on the canyon floor on a line directly between them. The angles of depression from each hiker to the landmark meter are 37\u00b0 and 21\u00b0. How far apart are the hikers? Round your […]<\/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-207486","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/207486","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=207486"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/207486\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=207486"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=207486"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=207486"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}