{"id":244562,"date":"2025-07-05T08:47:12","date_gmt":"2025-07-05T08:47:12","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=244562"},"modified":"2025-07-05T08:47:14","modified_gmt":"2025-07-05T08:47:14","slug":"make-a-polar-plot-of-the-function","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/05\/make-a-polar-plot-of-the-function\/","title":{"rendered":"Make a polar plot of the function"},"content":{"rendered":"\n
    \n
  1. Make a polar plot of the function: for
    The plot is shown on the right.<\/li>\n<\/ol>\n\n\n\n
    \"\"<\/figure>\n\n\n\n

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

    Correct Answer:<\/strong><\/p>\n\n\n\n

    The function is defined by the following equations for the polar coordinates (r, \u03b8):<\/p>\n\n\n\n

    \u03b8 = n * 135.7\u00b0
    r = \u221an
    for n = 1, 2, 3, …, 100<\/p>\n\n\n\n

    Explanation<\/h3>\n\n\n\n

    This problem illustrates how to generate a pattern, known as a phyllotactic or Fermat’s spiral, by plotting a series of points in a polar coordinate system. A polar system defines a point not by its x and y coordinates, but by its distance from the origin (radius, r) and its angle from a reference axis (\u03b8).<\/p>\n\n\n\n

    The provided function calculates the coordinates for 100 distinct points, corresponding to the integer values of n from 1 to 100. For each value of n, a unique pair of (r, \u03b8) coordinates is generated.<\/p>\n\n\n\n

      \n
    1. The Radius\u00a0r = \u221an<\/strong>: This equation determines the point’s distance from the center of the plot. As\u00a0n\u00a0increases, the radius\u00a0r\u00a0also increases, causing the points to move progressively farther away from the origin. However, because it’s a square root function, the distance between consecutive points decreases as\u00a0n\u00a0gets larger, making the spiral appear denser at its outer edges.<\/li>\n\n\n\n
    2. The Angle\u00a0\u03b8 = n * 135.7\u00b0<\/strong>: This equation sets the angle for each point. For every new point (n), the angle is increased by a constant amount of 135.7\u00b0. This fixed angular increment is called the divergence angle.<\/li>\n<\/ol>\n\n\n\n

      The combination of these two rules creates the striking spiral pattern. As points are plotted, the angle “wraps around” the 360\u00b0 circle. For example, the point for\u00a0n=3\u00a0is at an angle of\u00a03 * 135.7\u00b0 = 407.1\u00b0, which is equivalent to\u00a0407.1\u00b0 – 360\u00b0 = 47.1\u00b0. Because 135.7\u00b0 is an irrational-like fraction of 360\u00b0, the points never perfectly align, instead forming a set of interlocking spiral arms. This pattern is famously seen in nature in the arrangement of seeds in a sunflower, petals on a flower, and scales on a pinecone, as it represents a highly efficient way to pack elements.<\/p>\n\n\n\n

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

      The Correct Answer and Explanation is: Correct Answer: The function is defined by the following equations for the polar coordinates (r, \u03b8): \u03b8 = n * 135.7\u00b0r = \u221anfor n = 1, 2, 3, …, 100 Explanation This problem illustrates how to generate a pattern, known as a phyllotactic or Fermat’s spiral, by plotting a […]<\/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-244562","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/244562","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=244562"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/244562\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=244562"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=244562"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=244562"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}