{"id":185227,"date":"2025-01-22T09:17:22","date_gmt":"2025-01-22T09:17:22","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=185227"},"modified":"2025-01-22T09:17:24","modified_gmt":"2025-01-22T09:17:24","slug":"st-petersburg-russia-is-located-at-30-east-longitude-and-60-north-latitude-2","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-2\/","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. 6.0) 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

To solve this problem, we’ll convert the given geographical coordinates of St. Petersburg (latitude 60\u00b0N, longitude 30\u00b0E) into spherical and rectangular coordinates for a sphere with radius ( r = 4000 ) miles.<\/p>\n\n\n\n

\n\n\n\n

1. Spherical Coordinates<\/strong><\/h3>\n\n\n\n

Spherical coordinates ((\\rho, \\theta, \\phi)) are defined as:<\/p>\n\n\n\n

    \n
  • ( \\rho ): The radius of the sphere (4000 miles for Earth).<\/li>\n\n\n\n
  • ( \\theta ): The azimuthal angle measured from the positive x-axis in the xy-plane (longitude).<\/li>\n\n\n\n
  • ( \\phi ): The polar angle measured from the positive z-axis (co-latitude = ( 90^\\circ – \\text{latitude} )).<\/li>\n<\/ul>\n\n\n\n

    Given:<\/h4>\n\n\n\n
      \n
    • Latitude = ( 60^\\circ ), so co-latitude ( \\phi = 90^\\circ – 60^\\circ = 30^\\circ ).<\/li>\n\n\n\n
    • Longitude = ( 30^\\circ ), so ( \\theta = 30^\\circ ).<\/li>\n\n\n\n
    • Radius ( \\rho = 4000 ) miles.<\/li>\n<\/ul>\n\n\n\n

      Thus, the spherical coordinates<\/strong> are:
      [
      (\\rho, \\theta, \\phi) = (4000, 30^\\circ, 30^\\circ)
      ]<\/p>\n\n\n\n

      \n\n\n\n

      2. Rectangular Coordinates<\/strong><\/h3>\n\n\n\n

      Rectangular coordinates ((x, y, z)) are derived using:
      [
      x = \\rho \\sin\\phi \\cos\\theta
      ]
      [
      y = \\rho \\sin\\phi \\sin\\theta
      ]
      [
      z = \\rho \\cos\\phi
      ]<\/p>\n\n\n\n

      Calculations:<\/h4>\n\n\n\n
        \n
      • (\\phi = 30^\\circ), so (\\sin\\phi = \\frac{1}{2}), (\\cos\\phi = \\frac{\\sqrt{3}}{2}).<\/li>\n\n\n\n
      • (\\theta = 30^\\circ), so (\\cos\\theta = \\frac{\\sqrt{3}}{2}), (\\sin\\theta = \\frac{1}{2}).<\/li>\n\n\n\n
      • Substituting (\\rho = 4000):
        [
        x = 4000 \\cdot \\frac{1}{2} \\cdot \\frac{\\sqrt{3}}{2} = 1000\\sqrt{3} \\approx 1732 \\, \\text{miles}
        ]
        [
        y = 4000 \\cdot \\frac{1}{2} \\cdot \\frac{1}{2} = 1000 \\, \\text{miles}
        ]
        [
        z = 4000 \\cdot \\frac{\\sqrt{3}}{2} = 2000\\sqrt{3} \\approx 3464 \\, \\text{miles}
        ]<\/li>\n<\/ul>\n\n\n\n

        Thus, the rectangular coordinates<\/strong> are:
        [
        (x, y, z) \\approx (1732, 1000, 3464) \\, \\text{miles}.
        ]<\/p>\n\n\n\n

        \n\n\n\n

        Explanation (300 words)<\/strong><\/h3>\n\n\n\n

        St. Petersburg’s location on the globe can be expressed in both spherical and rectangular coordinate systems. Spherical coordinates are ideal for representing positions on a sphere, such as Earth, and are given by the radius, azimuthal angle (longitude), and polar angle (co-latitude). Rectangular coordinates, on the other hand, are Cartesian coordinates that represent a point in 3D space.<\/p>\n\n\n\n

        The spherical coordinates of St. Petersburg are derived directly from its geographical location. The Earth’s radius (( \\rho = 4000 ) miles) determines the radial distance. The azimuthal angle (( \\theta = 30^\\circ )) corresponds to its eastward position relative to the prime meridian. The polar angle (( \\phi = 30^\\circ )) is calculated as the complement of the latitude.<\/p>\n\n\n\n

        Rectangular coordinates are calculated by projecting the spherical coordinates onto the (x), (y), and (z) axes. These projections are determined using trigonometric relationships:<\/p>\n\n\n\n

          \n
        • (x) is the projection on the equatorial plane in the direction of the prime meridian.<\/li>\n\n\n\n
        • (y) is the projection perpendicular to the prime meridian.<\/li>\n\n\n\n
        • (z) is the projection along the Earth’s axis of rotation (northward).<\/li>\n<\/ul>\n\n\n\n

          These coordinates allow us to represent St. Petersburg\u2019s position in a mathematical model of Earth, enabling applications in fields like navigation, astronomy, and geophysics.<\/p>\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. 6.0) 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 Correct […]<\/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-185227","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/185227","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=185227"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/185227\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=185227"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=185227"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=185227"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}