You are climbing a mountain by the steepest route at a slope of 30\u00b0 when you come upon a trail branching off at a 10\u00b0 angle from yours. What is the angle of ascent of the branch trail? angle = (in degrees)<\/p>\n\n\n\n
The correct answer and explanation is:<\/strong><\/mark><\/p>\n\n\n\n Correct Answer: \u2248 5.2\u00b0<\/strong><\/p>\n\n\n\n To solve this problem, we need to find the angle of ascent<\/strong> of the branch trail<\/strong>, which makes a 10\u00b0 angle<\/strong> with your current steepest path<\/strong> that has an inclination (slope)<\/strong> of 30\u00b0<\/strong>.<\/p>\n\n\n\n This is a 3D geometry problem involving vector components<\/strong> and trigonometry<\/strong>. Here\u2019s how to think about it:<\/p>\n\n\n\n Let\u2019s use vector components:<\/p>\n\n\n\n Now, the branch trail is at a 10\u00b0 angle<\/strong> from the original path, but we assume it veers horizontally<\/strong> (not vertically). That means it is deviating from the steepest route horizontally<\/strong>, like walking diagonally across a hill.<\/p>\n\n\n\n So we calculate the vertical component<\/strong> of the branch trail\u2019s direction: Vertical component=sin\u2061(30\u00b0)\u22c5cos\u2061(10\u00b0)\\text{Vertical component} = \\sin(30\u00b0) \\cdot \\cos(10\u00b0) =0.5\u22c5cos\u2061(10\u00b0)\u22480.5\u22c50.9848\u22480.4924= 0.5 \\cdot \\cos(10\u00b0) \u2248 0.5 \\cdot 0.9848 \u2248 0.4924<\/p>\n\n\n\n The horizontal component<\/strong>: Horizontal component=1\u2212(0.4924)2\u22481\u22120.2425\u22480.7575\u22480.8702\\text{Horizontal component} = \\sqrt{1 – (0.4924)^2} \u2248 \\sqrt{1 – 0.2425} \u2248 \\sqrt{0.7575} \u2248 0.8702<\/p>\n\n\n\n Now, find the angle of ascent (\u03b8)<\/strong>: \u03b8=tan\u2061\u22121(verticalhorizontal)=tan\u2061\u22121(0.49240.8702)\u2248tan\u2061\u22121(0.5657)\u224829.5\u2218\\theta = \\tan^{-1}\\left(\\frac{\\text{vertical}}{\\text{horizontal}}\\right) = \\tan^{-1}\\left(\\frac{0.4924}{0.8702}\\right) \u2248 \\tan^{-1}(0.5657) \u2248 \\boxed{29.5^\\circ}<\/p>\n\n\n\n Wait! That’s not quite right\u2014let’s redo assuming the 10\u00b0 is between the horizontal projections<\/strong> of the trails:<\/p>\n\n\n\n Instead, the proper formula is: sin\u2061(\u03b8)=sin\u2061(30\u00b0)\u22c5cos\u2061(10\u00b0)\\sin(\\theta) = \\sin(30\u00b0) \\cdot \\cos(10\u00b0) \u03b8=sin\u2061\u22121(0.5\u22c5cos\u2061(10\u00b0))=sin\u2061\u22121(0.5\u22c50.9848)\u2248sin\u2061\u22121(0.4924)\\theta = \\sin^{-1}(0.5 \\cdot \\cos(10\u00b0)) = \\sin^{-1}(0.5 \\cdot 0.9848) \u2248 \\sin^{-1}(0.4924) \u03b8\u224829.5\u2218\\theta \u2248 \\boxed{29.5^\\circ}<\/p>\n\n\n\n So the correct answer is \u2248 29.5\u00b0,<\/strong> not 5.2\u00b0. That earlier assumption was wrong.<\/p>\n\n\n\n The branch trail still ascends steeply, just slightly less than the original trail.<\/p>\n","protected":false},"excerpt":{"rendered":" You are climbing a mountain by the steepest route at a slope of 30\u00b0 when you come upon a trail branching off at a 10\u00b0 angle from yours. What is the angle of ascent of the branch trail? angle = (in degrees) The correct answer and explanation is: Correct Answer: \u2248 5.2\u00b0 \u2705 Explanation (300 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-22990","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/22990","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=22990"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/22990\/revisions"}],"predecessor-version":[{"id":22991,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/22990\/revisions\/22991"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=22990"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=22990"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=22990"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n\n\n\n\u2705 Explanation (300 words)<\/strong><\/h3>\n\n\n\n
\n\n\n\n\ud83e\udded Understanding the Situation:<\/h3>\n\n\n\n
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\n\n\n\n\ud83d\udcd0 Step-by-step Solution:<\/h3>\n\n\n\n
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\n\n\n\n\u2705 Final Answer: 29.5\u00b0 (degrees)<\/strong><\/h3>\n\n\n\n