{"id":185224,"date":"2025-01-22T09:14:12","date_gmt":"2025-01-22T09:14:12","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=185224"},"modified":"2025-01-22T09:14:13","modified_gmt":"2025-01-22T09:14:13","slug":"st-petersburg-russia-is-located-at-30-east-longitude-and-60-north-latitude","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/01\/22\/st-petersburg-russia-is-located-at-30-east-longitude-and-60-north-latitude\/","title":{"rendered":"St. Petersburg, Russia, is located at 30\u00b0 east longitude and 60\u00b0 north latitude"},"content":{"rendered":"\n

St. Petersburg, Russia, is located at 30\u00b0 east longitude and 60\u00b0 north latitude. Find the spherical coordinates (p, 6, 8) and the rectangular coordinates (x, y, z) relative to a globe with poles on the z-axis and with prime meridian in the zz-plane. Assume that the Earth is a sphere of radius 4000 miles.<\/p>\n\n\n\n

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

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

Correct Answer:<\/h3>\n\n\n\n

Spherical Coordinates:<\/h4>\n\n\n\n

[
(p, \\theta, \\phi) = (4000, 0.524 \\, \\text{radians}, 0.524 \\, \\text{radians})
]<\/p>\n\n\n\n

Rectangular Coordinates:<\/h4>\n\n\n\n

[
(x, y, z) = (1732.05, 1000.00, 3464.10) \\, \\text{miles}
]<\/p>\n\n\n\n

\n\n\n\n

Explanation (300 Words):<\/h3>\n\n\n\n

The problem involves converting geographical coordinates into spherical and rectangular coordinates, assuming the Earth is a perfect sphere of radius 4000 miles.<\/p>\n\n\n\n

    \n
  1. Given Data:<\/strong><\/li>\n<\/ol>\n\n\n\n
      \n
    • Latitude ((60^\\circ))<\/strong>: North latitude indicates the angle from the equatorial plane.<\/li>\n\n\n\n
    • Longitude ((30^\\circ))<\/strong>: East longitude specifies the angle from the prime meridian.<\/li>\n\n\n\n
    • Earth’s radius ((p = 4000) miles).<\/li>\n<\/ul>\n\n\n\n
        \n
      1. Conversion to Spherical Coordinates ((p, \\theta, \\phi)):<\/strong><\/li>\n<\/ol>\n\n\n\n
          \n
        • (p) is the Earth’s radius, given as 4000 miles.<\/li>\n\n\n\n
        • The longitude ((30^\\circ)) is converted to radians: (\\theta = \\frac{\\pi}{6} \\approx 0.524 \\, \\text{radians}).<\/li>\n\n\n\n
        • Latitude is converted to colatitude ((\\phi)): (\\phi = 90^\\circ – 60^\\circ = 30^\\circ), which is (\\frac{\\pi}{6} \\approx 0.524 \\, \\text{radians}).<\/li>\n<\/ul>\n\n\n\n
            \n
          1. Rectangular Coordinates ((x, y, z)):<\/strong><\/li>\n<\/ol>\n\n\n\n
              \n
            • Use spherical-to-rectangular formulas:
              [
              x = p \\cdot \\sin(\\phi) \\cdot \\cos(\\theta)
              ]
              [
              y = p \\cdot \\sin(\\phi) \\cdot \\sin(\\theta)
              ]
              [
              z = p \\cdot \\cos(\\phi)
              ]<\/li>\n\n\n\n
            • Substituting ((p = 4000, \\phi \\approx 0.524, \\theta \\approx 0.524)):\n
                \n
              • (x \\approx 1732.05) miles<\/li>\n\n\n\n
              • (y \\approx 1000.00) miles<\/li>\n\n\n\n
              • (z \\approx 3464.10) miles<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n
                  \n
                1. Interpretation:<\/strong><\/li>\n<\/ol>\n\n\n\n
                    \n
                  • The spherical coordinates specify the position using radius, longitude, and colatitude.<\/li>\n\n\n\n
                  • The rectangular coordinates ((x, y, z)) describe the same position in a Cartesian system, centered at the Earth’s core with the (z)-axis aligned with the poles.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"

                    St. Petersburg, Russia, is located at 30\u00b0 east longitude and 60\u00b0 north latitude. Find the spherical coordinates (p, 6, 8) and the rectangular coordinates (x, y, z) relative to a globe with poles on the z-axis and with prime meridian in the zz-plane. Assume that the Earth is a sphere of radius 4000 miles. The […]<\/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-185224","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/185224","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=185224"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/185224\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=185224"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=185224"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=185224"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}