{"id":237668,"date":"2025-06-17T11:56:42","date_gmt":"2025-06-17T11:56:42","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=237668"},"modified":"2025-06-17T11:56:44","modified_gmt":"2025-06-17T11:56:44","slug":"choose__-choose-from-0-to-360-degrees-from-0-to-90-degrees-0-3926990818-1-570796327","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/17\/choose__-choose-from-0-to-360-degrees-from-0-to-90-degrees-0-3926990818-1-570796327\/","title":{"rendered":"Choose__ Choose from 0 to 360 degrees from 0 to 90 degrees 0.3926990818 1.570796327"},"content":{"rendered":"\n

Placed in the first octant of 3-dimensional (3D) coordinates. The equation below is given by 2x^2 + y^2 = 1. The double integral \u00e2\u02c6\u00ab\u00e2\u02c6\u00abf(r, \u00ce\u00b8) r dr d\u00ce\u00b8 value of \u00ce\u00b8? Choose. Name of the gemstone? Choose__ Choose from 0 to 360 degrees from 0 to 90 degrees 0.3926990818 1.570796327<\/p>\n\n\n\n

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

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

To determine the correct interval for \u03b8\\theta, consider the given equation 2×2+y2=12x^2 + y^2 = 1, which represents an elliptical region in the first octant.<\/p>\n\n\n\n

Step 1: Convert to Polar Coordinates<\/h3>\n\n\n\n

The transformation equations are:<\/p>\n\n\n\n

x=rcos\u2061\u03b8,y=rsin\u2061\u03b8x = r\\cos\\theta, \\quad y = r\\sin\\theta<\/p>\n\n\n\n

Substituting into the given equation:<\/p>\n\n\n\n

2(rcos\u2061\u03b8)2+(rsin\u2061\u03b8)2=12(r\\cos\\theta)^2 + (r\\sin\\theta)^2 = 1<\/p>\n\n\n\n

2r2cos\u20612\u03b8+r2sin\u20612\u03b8=12r^2\\cos^2\\theta + r^2\\sin^2\\theta = 1<\/p>\n\n\n\n

Factor out r2r^2:<\/p>\n\n\n\n

r2(2cos\u20612\u03b8+sin\u20612\u03b8)=1r^2(2\\cos^2\\theta + \\sin^2\\theta) = 1<\/p>\n\n\n\n

Solving for r2r^2:<\/p>\n\n\n\n

r2=12cos\u20612\u03b8+sin\u20612\u03b8r^2 = \\frac{1}{2\\cos^2\\theta + \\sin^2\\theta}<\/p>\n\n\n\n

Step 2: Identify the Range of \u03b8\\theta<\/h3>\n\n\n\n

The constraint that the region is in the first octant<\/strong> means:<\/p>\n\n\n\n

    \n
  • x\u22650x \\geq 0 and y\u22650y \\geq 0<\/li>\n\n\n\n
  • \u03b8\\theta spans from the positive x-axis to the positive y-axis<\/li>\n<\/ul>\n\n\n\n

    In polar coordinates, this means \u03b8\\theta ranges from 00 to 90\u221890^\\circ (\u03c02\\frac{\\pi}{2} radians).<\/p>\n\n\n\n

    Step 3: Choose the Correct Value<\/h3>\n\n\n\n

    Among the options provided:<\/p>\n\n\n\n

      \n
    • 0.39269908180.3926990818 radians \u224822.5\u2218\\approx 22.5^\\circ<\/li>\n\n\n\n
    • 1.5707963271.570796327 radians \u224890\u2218\\approx 90^\\circ<\/li>\n<\/ul>\n\n\n\n

      Since the first octant spans up to 90\u221890^\\circ, the correct choice is 1.5707963271.570796327 radians.<\/p>\n\n\n\n

      Step 4: Gemstone Shape and Interpretation<\/h3>\n\n\n\n

      The integral likely represents the volume<\/strong> of a gemstone-like solid constrained within the first octant. Given the elliptical base and the integral setup, it resembles a modified ellipsoid or paraboloid fragment<\/strong>.<\/p>\n\n\n\n

      The correct interpretation ensures the choice aligns with first-octant constraints, confirming \u03b8\\theta spans from 00 to 90\u221890^\\circ.<\/p>\n\n\n\n

      \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

      Placed in the first octant of 3-dimensional (3D) coordinates. The equation below is given by 2x^2 + y^2 = 1. The double integral \u00e2\u02c6\u00ab\u00e2\u02c6\u00abf(r, \u00ce\u00b8) r dr d\u00ce\u00b8 value of \u00ce\u00b8? Choose. Name of the gemstone? Choose__ Choose from 0 to 360 degrees from 0 to 90 degrees 0.3926990818 1.570796327 The Correct Answer and Explanation […]<\/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-237668","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/237668","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=237668"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/237668\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=237668"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=237668"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=237668"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}