A hiker starts her trip by walking 25.0 km southeast from her car and sets up her tent for the night.<\/p>\n\n\n\n
On the second day, she continues her journey by walking 40.0 km in a direction 60\u00b0 north of east, where she eventually finds a forest ranger’s tower.<\/p>\n\n\n\n
Determine the hiker’s final position relative to her starting point (car).<\/p>\n\n\n\n
The correct answer and explanation is :<\/strong><\/mark><\/p>\n\n\n\n To solve this problem, we need to determine the hiker’s final position relative to her starting point (the car) after two days of walking. The key here is to break the movements into components along the east and north directions (using Cartesian coordinates), then combine them to find the final position.<\/p>\n\n\n\n The hiker walks 25.0 km southeast. Southeast corresponds to an angle of 45\u00b0 south of east. This means that the hiker is moving both in the eastward (x) and southward (y) directions. We can break this movement into two components using trigonometry:<\/p>\n\n\n\n Thus, after day 1, the hiker has moved 17.68 km east and 17.68 km south.<\/p>\n\n\n\n On the second day, the hiker walks 40.0 km in a direction 60\u00b0 north of east. We can break this movement into eastward (x) and northward (y) components:<\/p>\n\n\n\n Thus, after day 2, the hiker has moved 20.0 km east and 34.64 km north.<\/p>\n\n\n\n To find the final position, we add up the components from both days:<\/p>\n\n\n\n Thus, the hiker’s final position relative to the car is 37.68 km east and 16.96 km north. The final displacement (straight-line distance from the starting point) can be calculated using the Pythagorean theorem:<\/p>\n\n\n\n [ Thus, the hiker’s final position relative to the starting point is approximately 41.3 km<\/strong> at an angle north of east. The angle can be calculated using:<\/p>\n\n\n\n [ Hence, the hiker’s final position is 41.3 km<\/strong> at 24.4\u00b0 north of east<\/strong> relative to her starting point.<\/p>\n","protected":false},"excerpt":{"rendered":" A hiker starts her trip by walking 25.0 km southeast from her car and sets up her tent for the night. On the second day, she continues her journey by walking 40.0 km in a direction 60\u00b0 north of east, where she eventually finds a forest ranger’s tower. Determine the hiker’s final position relative to […]<\/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-200069","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/200069","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=200069"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/200069\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=200069"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=200069"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=200069"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Day 1:<\/h3>\n\n\n\n
\n
( x\u2081 = 25.0 \\, \\text{km} \\times \\cos(45^\\circ) = 25.0 \\, \\text{km} \\times 0.7071 \\approx 17.68 \\, \\text{km} )<\/li>\n\n\n\n
( y\u2081 = 25.0 \\, \\text{km} \\times \\sin(45^\\circ) = 25.0 \\, \\text{km} \\times 0.7071 \\approx 17.68 \\, \\text{km} )<\/li>\n<\/ul>\n\n\n\nDay 2:<\/h3>\n\n\n\n
\n
( x\u2082 = 40.0 \\, \\text{km} \\times \\cos(60^\\circ) = 40.0 \\, \\text{km} \\times 0.5 = 20.0 \\, \\text{km} )<\/li>\n\n\n\n
( y\u2082 = 40.0 \\, \\text{km} \\times \\sin(60^\\circ) = 40.0 \\, \\text{km} \\times 0.866 \\approx 34.64 \\, \\text{km} )<\/li>\n<\/ul>\n\n\n\nFinal Position:<\/h3>\n\n\n\n
\n
( x_{\\text{final}} = x\u2081 + x\u2082 = 17.68 \\, \\text{km} + 20.0 \\, \\text{km} = 37.68 \\, \\text{km} )<\/li>\n\n\n\n
( y_{\\text{final}} = y\u2082 – y\u2081 = 34.64 \\, \\text{km} – 17.68 \\, \\text{km} = 16.96 \\, \\text{km} )<\/li>\n<\/ul>\n\n\n\n
\\text{Displacement} = \\sqrt{x_{\\text{final}}^2 + y_{\\text{final}}^2} = \\sqrt{(37.68)^2 + (16.96)^2} \\approx \\sqrt{1417.1 + 287.4} \\approx \\sqrt{1704.5} \\approx 41.3 \\, \\text{km}
]<\/p>\n\n\n\n
\\theta = \\tan^{-1}\\left(\\frac{y_{\\text{final}}}{x_{\\text{final}}}\\right) = \\tan^{-1}\\left(\\frac{16.96}{37.68}\\right) \\approx \\tan^{-1}(0.45) \\approx 24.4^\\circ
]<\/p>\n\n\n\n