{"id":1289,"date":"2026-01-26T18:39:16","date_gmt":"2026-01-26T08:39:16","guid":{"rendered":"https:\/\/archive4ones.com\/2ndstudy\/?p=1289"},"modified":"2026-01-27T12:19:35","modified_gmt":"2026-01-27T02:19:35","slug":"year12-math-1-2-24-earth-geometry","status":"publish","type":"post","link":"https:\/\/archive4ones.com\/2ndstudy\/?p=1289","title":{"rendered":"Year12-MATH-2-4-3 Earth geometry"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">In the&nbsp;<strong>QLD Year 12 General Mathematics<\/strong>&nbsp;syllabus, Unit 3 is titled &#8220;<mark><strong>Bivariate data, sequences and change, and Earth geometry<\/strong><\/mark>&#8220;. This unit covers data analysis (bivariate and time series), mathematical sequences (arithmetic and geometric), and the spatial measurement of the globe.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Earth Geometry and Time Zones<\/strong>&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This topic applies spherical geometry to the Earth to calculate distances and determine time differences.&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Positioning:<\/strong> Students use <strong>latitude<\/strong> (north-south) and <strong>longitude<\/strong> (east-west) in degrees and minutes to locate positions relative to the Equator and Prime Meridian.<\/li>\n\n\n\n<li><strong>Spherical Concepts:<\/strong> Calculations involve <strong>great circles<\/strong> (circles that share the Earth&#8217;s center, such as the Equator or meridians) and <strong>small circles<\/strong> (parallels of latitude other than the Equator).<\/li>\n\n\n\n<li><strong>Distance Calculations:<\/strong>\n<ul class=\"wp-block-list\">\n<li><strong>Along a Meridian (Great Circle):<\/strong> <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>D<\/mi><mo>=<\/mo><mn>111.2<\/mn><mo>\u00d7<\/mo><mtext>angular distance<\/mtext><\/mrow><annotation encoding=\"text\/plain\">cap D equals 111.2 cross angular distance<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li><strong>Along a Parallel of Latitude (Small Circle):<\/strong> <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>D<\/mi><mo>=<\/mo><mn>111.2<\/mn><mi>cos<\/mi><mo>(<\/mo><mi>\u03b8<\/mi><mo>)<\/mo><mo>\u00d7<\/mo><mtext>angular distance<\/mtext><\/mrow><annotation encoding=\"text\/plain\">cap D equals 111.2 cosine open paren theta close paren cross angular distance<\/annotation><\/semantics><\/math>, where \ud835\udf03 is the latitude.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Time Zones:<\/strong> Calculations are based on longitude differences, where <math data-latex=\" 15^{\\circ }\"><semantics><msup><mn>15<\/mn><mo lspace=\"0em\" rspace=\"0em\">\u2218<\/mo><\/msup><annotation encoding=\"application\/x-tex\"> 15^{\\circ }<\/annotation><\/semantics><\/math><strong> equals 1 hour<\/strong> of time difference.&nbsp;<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Arithmetic and Geometric Sequences<\/strong>&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This section focuses on patterns of growth and decay using recursion and general rules.&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Arithmetic Sequences:<\/strong> These have a <strong>common difference<\/strong> (\ud835\udc51).\n<ul class=\"wp-block-list\">\n<li><strong>Recursive Rule:<\/strong> <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>t<\/mi><mrow><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo>=<\/mo><msub><mi>t<\/mi><mi>n<\/mi><\/msub><mo>+<\/mo><mi>d<\/mi><\/mrow><annotation encoding=\"text\/plain\">t sub n plus 1 end-sub equals t sub n plus d<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li><strong>General Rule (<\/strong><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><msup><mi>n<\/mi><mrow><mi>t<\/mi><mi>h<\/mi><\/mrow><\/msup><annotation encoding=\"text\/plain\">n raised to the t h power<\/annotation><\/semantics><\/math>term<strong>):<\/strong> <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>t<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><mi>a<\/mi><mo>+<\/mo><mo>(<\/mo><mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>)<\/mo><mi>d<\/mi><\/mrow><annotation encoding=\"text\/plain\">t sub n equals a plus open paren n minus 1 close paren d<\/annotation><\/semantics><\/math>, where \ud835\udc4e is the first term.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Geometric Sequences:<\/strong> These involve a <strong>common ratio<\/strong> (\ud835\udc5f), used to model exponential growth or decay (e.g., population growth or reducing-balance depreciation).\n<ul class=\"wp-block-list\">\n<li><strong>Recursive Rule:<\/strong> <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>t<\/mi><mrow><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo>=<\/mo><mi>r<\/mi><mo>\u00d7<\/mo><msub><mi>t<\/mi><mi>n<\/mi><\/msub><\/mrow><annotation encoding=\"text\/plain\">t sub n plus 1 end-sub equals r cross t sub n<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li><strong>General Rule (<\/strong><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><msup><mi>n<\/mi><mrow><mi>t<\/mi><mi>h<\/mi><\/mrow><\/msup><annotation encoding=\"text\/plain\">n raised to the t h power<\/annotation><\/semantics><\/math><strong>term):<\/strong> <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>t<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><mi>a<\/mi><mo>\u00d7<\/mo><msup><mi>r<\/mi><mrow><mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><annotation encoding=\"text\/plain\">t sub n equals a cross r raised to the n minus 1 power<\/annotation><\/semantics><\/math>.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">*****************************************************************************************<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">QLD 12\u5e74\u751f\u4e00\u822c\u6570\u5b66\u30b7\u30e9\u30d0\u30b9\u306e\u30e6\u30cb\u30c3\u30c83\u306f\u3001\u300c\u4e8c\u5909\u91cf\u30c7\u30fc\u30bf\u3001\u6570\u5217\u3068\u5909\u5316\u3001\u305d\u3057\u3066\u5730\u7403\u5e7e\u4f55\u5b66\u300d\u3067\u3059\u3002\u3053\u306e\u30e6\u30cb\u30c3\u30c8\u3067\u306f\u3001\u30c7\u30fc\u30bf\u5206\u6790\uff08\u4e8c\u5909\u91cf\u304a\u3088\u3073\u6642\u7cfb\u5217\uff09\u3001\u6570\u5217\uff08\u7b97\u8853\u304a\u3088\u3073\u5e7e\u4f55\uff09\u3001\u305d\u3057\u3066\u5730\u7403\u306e\u7a7a\u9593\u6e2c\u5b9a\u306b\u3064\u3044\u3066\u5b66\u3073\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u5730\u7403\u5e7e\u4f55\u5b66\u3068\u30bf\u30a4\u30e0\u30be\u30fc\u30f3<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u306e\u30c8\u30d4\u30c3\u30af\u3067\u306f\u3001\u7403\u9762\u5e7e\u4f55\u5b66\u3092\u5730\u7403\u306b\u9069\u7528\u3057\u3001\u8ddd\u96e2\u306e\u8a08\u7b97\u3068\u6642\u5dee\u306e\u5224\u5b9a\u3092\u884c\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u4f4d\u7f6e\uff1a\u751f\u5f92\u306f\u7def\u5ea6\uff08\u5357\u5317\uff09\u3068\u7d4c\u5ea6\uff08\u6771\u897f\uff09\u3092\u5ea6\u3068\u5206\u3067\u8868\u3057\u3001\u8d64\u9053\u3068\u672c\u521d\u5b50\u5348\u7dda\u3092\u57fa\u6e96\u3068\u3057\u305f\u4f4d\u7f6e\u3092\u7279\u5b9a\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u7403\u9762\u6982\u5ff5\uff1a\u8a08\u7b97\u306b\u306f\u3001\u5927\u5186\uff08\u8d64\u9053\u3084\u5b50\u5348\u7dda\u306a\u3069\u3001\u5730\u7403\u306e\u4e2d\u5fc3\u3092\u5171\u6709\u3059\u308b\u5186\uff09\u3068\u5c0f\u5186\uff08\u8d64\u9053\u4ee5\u5916\u306e\u7def\u7dda\uff09\u304c\u7528\u3044\u3089\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u8ddd\u96e2\u8a08\u7b97\uff1a<br>\u5b50\u5348\u7dda\uff08\u5927\u5186\uff09\u6cbf\u3044\uff1a(<math data-latex=\"D=111.2\\times \\text{\u89d2\u5ea6\\\u8ddd\u96e2}\"><semantics><mrow><mi>D<\/mi><mo>=<\/mo><mn>111.2<\/mn><mo>\u00d7<\/mo><mrow><mtext>\u89d2\u5ea6<\/mtext><mtext style=\"color:#b22222;\">\\\u8ddd<\/mtext><mtext>\u96e2<\/mtext><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">D=111.2\\times \\text{\u89d2\u5ea6\\\u8ddd\u96e2}<\/annotation><\/semantics><\/math>)\u3002<br>\u7def\u7dda\uff08\u5c0f\u5186\uff09\u6cbf\u3044\uff1a(<math data-latex=\"D=111.2\\cos (\\theta )\\times \\text{\u89d2\u5ea6\\\u8ddd\u96e2}\"><semantics><mrow><mi>D<\/mi><mo>=<\/mo><mn>111.2<\/mn><mrow><mspace width=\"0.1667em\"><\/mspace><mi>cos<\/mi><mo>\u2061<\/mo><\/mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>\u03b8<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u00d7<\/mo><mrow><mtext>\u89d2\u5ea6<\/mtext><mtext style=\"color:#b22222;\">\\\u8ddd<\/mtext><mtext>\u96e2<\/mtext><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">D=111.2\\cos (\\theta )\\times \\text{\u89d2\u5ea6\\\u8ddd\u96e2}<\/annotation><\/semantics><\/math>)\u3002\u3053\u3053\u3067(<math data-latex=\"\\theta \"><semantics><mi>\u03b8<\/mi><annotation encoding=\"application\/x-tex\">\\theta <\/annotation><\/semantics><\/math>)\u306f\u7def\u5ea6\u3067\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u30bf\u30a4\u30e0\u30be\u30fc\u30f3\uff1a\u8a08\u7b97\u306f\u7d4c\u5ea6\u306e\u5dee\u306b\u57fa\u3065\u3044\u3066\u884c\u308f\u308c\u307e\u3059\u3002(<math data-latex=\"15^{\\circ }\"><semantics><msup><mn>15<\/mn><mo lspace=\"0em\" rspace=\"0em\">\u2218<\/mo><\/msup><annotation encoding=\"application\/x-tex\">15^{\\circ }<\/annotation><\/semantics><\/math>)\u306f1\u6642\u9593\u306e\u6642\u5dee\u306b\u76f8\u5f53\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u7b49\u5dee\u6570\u5217\u3068\u5e7e\u4f55\u6570\u5217<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u306e\u30bb\u30af\u30b7\u30e7\u30f3\u3067\u306f\u3001\u518d\u5e30\u3068\u4e00\u822c\u5247\u3092\u7528\u3044\u3066\u3001\u5897\u52a0\u3068\u6e1b\u5c11\u306e\u30d1\u30bf\u30fc\u30f3\u306b\u7126\u70b9\u3092\u5f53\u3066\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u7b49\u5dee\u6570\u5217\uff1a\u3053\u308c\u3089\u306f\u516c\u5dee(<math data-latex=\"d\"><semantics><mi>d<\/mi><annotation encoding=\"application\/x-tex\">d<\/annotation><\/semantics><\/math>)\u3092\u6301\u3061\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u518d\u5e30\u5247\uff1a(<math data-latex=\"t_{n+1}=t_{n}+d\"><semantics><mrow><msub><mi>t<\/mi><mrow><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo>=<\/mo><msub><mi>t<\/mi><mi>n<\/mi><\/msub><mo>+<\/mo><mi>d<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">t_{n+1}=t_{n}+d<\/annotation><\/semantics><\/math>)\u3002<br>\u4e00\u822c\u5247\uff08n\u756a\u76ee\u306e\u9805\uff09\uff1a<math data-latex=\"t_{n}=a+(n-1)d\"><semantics><mrow><msub><mi>t<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><mi>a<\/mi><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mi>d<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">t_{n}=a+(n-1)d<\/annotation><\/semantics><\/math>\uff08<math data-latex=\"a\"><semantics><mi>a<\/mi><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math>\uff09\u306f\u6700\u521d\u306e\u9805\u3067\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u7b49\u6bd4\u6570\u5217\uff1a\u516c\u6bd4\uff08r\\\uff09\u3092\u4f34\u3044\u3001\u6307\u6570\u95a2\u6570\u7684\u5897\u52a0\u307e\u305f\u306f\u6e1b\u5c11\uff08\u4f8b\uff1a\u4eba\u53e3\u5897\u52a0\u3084\u6e1b\u4fa1\u511f\u5374\uff09\u3092\u30e2\u30c7\u30eb\u5316\u3059\u308b\u305f\u3081\u306b\u4f7f\u7528\u3055\u308c\u307e\u3059\u3002<br>\u518d\u5e30\u7684\u898f\u5247\uff1a<math data-latex=\"t_{n+1}=r\\times t_{n}\"><semantics><mrow><msub><mi>t<\/mi><mrow><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo>=<\/mo><mi>r<\/mi><mo>\u00d7<\/mo><msub><mi>t<\/mi><mi>n<\/mi><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">t_{n+1}=r\\times t_{n}<\/annotation><\/semantics><\/math><br>\u4e00\u822c\u5247\uff08<math data-latex=\"n\"><semantics><mi>n<\/mi><annotation encoding=\"application\/x-tex\">n<\/annotation><\/semantics><\/math>\u756a\u76ee\u306e\u9805\uff09\uff1a<math data-latex=\"t_{n}=a\\times r^{n-1}\"><semantics><mrow><msub><mi>t<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><mi>a<\/mi><mo>\u00d7<\/mo><msup><mi>r<\/mi><mrow><mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">t_{n}=a\\times r^{n-1}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>In the&nbsp;QLD Year 12 General Mathematics&nbsp;syllabus, Unit 3 is titled &#8220;Bivariate data, sequences and change, and Earth geometry&#8220;. This unit covers data analysis (bivariate and time series), mathematical sequences (arithmetic and geometric), and the spatial measurement of the globe.&nbsp; Earth Geometry and Time Zones&nbsp; This topic applies spherical geometry to the Earth to calculate distances [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[8],"tags":[],"class_list":["post-1289","post","type-post","status-publish","format-standard","hentry","category-math"],"_links":{"self":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1289","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1289"}],"version-history":[{"count":4,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1289\/revisions"}],"predecessor-version":[{"id":1367,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1289\/revisions\/1367"}],"wp:attachment":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1289"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1289"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1289"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}