{"id":585,"date":"2009-07-10T22:45:05","date_gmt":"2009-07-10T21:45:05","guid":{"rendered":"http:\/\/www.fiveoclock.de\/?p=585"},"modified":"2009-07-10T22:47:06","modified_gmt":"2009-07-10T21:47:06","slug":"seilbahn","status":"publish","type":"post","link":"https:\/\/www.fiveoclock.de\/?p=585","title":{"rendered":"Seilbahnen"},"content":{"rendered":"<p>Nach einiger Zeit hier mal wieder ein R\u00e4tsel, auch aus <a  href=\"http:\/\/www.amazon.de\/gp\/product\/1846680646?ie=UTF8&#038;tag=fiocl-21&#038;linkCode=as2&#038;camp=1638&#038;creative=19454&#038;creativeASIN=1846680646\">Professor Stewart&#8217;s Cabinet of Mathematical Curiosities S.124<\/a><img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.assoc-amazon.de\/e\/ir?t=fiocl-21&#038;l=as2&#038;o=3&#038;a=1846680646\" width=\"1\" height=\"1\" border=\"0\" alt=\"\" style=\"border:none !important; margin:0px !important;\" \/>:<\/p>\n<p><img decoding=\"async\" src=\"http:\/\/www.fiveoclock.de\/wordpress\/wp-content\/uploads\/2009\/07\/seilbahn11.png\" alt=\"Seilbahn\" \/><\/p>\n<p>In einem Bergtal in dem rechts und links steile Klippen aufragen gibt es zwei Seilbahnen, eine geht vom linken Talpunkt <i>A<\/i> zur rechten Klippe <i>D<\/i> die sich auf 400 Metern H\u00f6he befindet und die andere geht vom rechten Talpunkt <i>C<\/i> zur linken Klippe <i>B<\/i> die sich auf 600 Metern H\u00f6he befindet. Auf welcher H\u00f6he <i>h<\/i> begegnen sich die Seilbahnen?<\/p>\n<p><a class=\"spoiler_link_show\" href=\"javascript:void(0)\" onclick=\"wpSpoilerToggle(document.getElementById('id1649039204'), this, 'Hinweis', 'Verstecken')\">Hinweis<\/a>\n<div class=\"spoiler_div\" id=\"id1649039204\" style=\"display:none\"><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/www.fiveoclock.de\/wordpress\/wp-content\/uploads\/2009\/07\/seilbahn2.png\" alt=\"Seilbahn\" \/><\/p>\n<p>In obiger Zeichnung sind Winkel eingezeichnet. Welche sind gleich und was kann man daraus schlie\u00dfen?<\/p>\n<p><\/div>\n<\/p>\n<p><a class=\"spoiler_link_show\" href=\"javascript:void(0)\" onclick=\"wpSpoilerToggle(document.getElementById('id1415030586'), this, 'L\u00f6sung', 'Ausblenden')\">L\u00f6sung<\/a>\n<div class=\"spoiler_div\" id=\"id1415030586\" style=\"display:none\"><br \/>\n<img decoding=\"async\" src=\"http:\/\/www.fiveoclock.de\/wordpress\/wp-content\/uploads\/2009\/07\/seilbahn3.png\" alt=\"Seilbahn\" \/><\/p>\n<p>Da folgende Winkel gleich sind: <span class=\"tex\"><img decoding=\"async\" src=\"\/tex?x=%5Calpha%20%3D%20%5Cbeta&#038;fg=00000&#038;s=0\" title=\"\\alpha = \\beta\" style=\"vertical-align:-20%;\" class=\"tex\" alt=\"\\alpha = \\beta\" \/><\/span>, <span class=\"tex\"><img decoding=\"async\" src=\"\/tex?x=%5Cgamma%20%3D%20%5Cdelta&#038;fg=00000&#038;s=0\" title=\"\\gamma = \\delta\" style=\"vertical-align:-20%;\" class=\"tex\" alt=\"\\gamma = \\delta\" \/><\/span> und <span class=\"tex\"><img decoding=\"async\" src=\"\/tex?x=%5Cepsilon%20%3D%20%5Czeta&#038;fg=00000&#038;s=0\" title=\"\\epsilon = \\zeta\" style=\"vertical-align:-20%;\" class=\"tex\" alt=\"\\epsilon = \\zeta\" \/><\/span>, sind die beiden Dreiecke<br \/>\n<span class=\"tex\"><img decoding=\"async\" src=\"\/tex?x=AJB&#038;fg=00000&#038;s=0\" title=\"AJB\" style=\"vertical-align:-20%;\" class=\"tex\" alt=\"AJB\" \/><\/span> und <span class=\"tex\"><img decoding=\"async\" src=\"\/tex?x=DJC&#038;fg=00000&#038;s=0\" title=\"DJC\" style=\"vertical-align:-20%;\" class=\"tex\" alt=\"DJC\" \/><\/span> \u00e4hnlich. Dreht man das Dreieck <span class=\"tex\"><img decoding=\"async\" src=\"\/tex?x=DJC&#038;fg=00000&#038;s=0\" title=\"DJC\" style=\"vertical-align:-20%;\" class=\"tex\" alt=\"DJC\" \/><\/span> um <span class=\"tex\"><img decoding=\"async\" src=\"\/tex?x=J&#038;fg=00000&#038;s=0\" title=\"J\" style=\"vertical-align:-20%;\" class=\"tex\" alt=\"J\" \/><\/span>, so erh\u00e4lt man <span class=\"tex\"><img decoding=\"async\" src=\"\/tex?x=D%27JC%27&#038;fg=00000&#038;s=0\" title=\"D'JC'\" style=\"vertical-align:-20%;\" class=\"tex\" alt=\"D'JC'\" \/><\/span>. Durch ein Hilfslinie parallel zum Boden durch <span class=\"tex\"><img decoding=\"async\" src=\"\/tex?x=J&#038;fg=00000&#038;s=0\" title=\"J\" style=\"vertical-align:-20%;\" class=\"tex\" alt=\"J\" \/><\/span> erh\u00e4lt man noch die Punkte <span class=\"tex\"><img decoding=\"async\" src=\"\/tex?x=G&#038;fg=00000&#038;s=0\" title=\"G\" style=\"vertical-align:-20%;\" class=\"tex\" alt=\"G\" \/><\/span>, <span class=\"tex\"><img decoding=\"async\" src=\"\/tex?x=H&#038;fg=00000&#038;s=0\" title=\"H\" style=\"vertical-align:-20%;\" class=\"tex\" alt=\"H\" \/><\/span> und <span class=\"tex\"><img decoding=\"async\" src=\"\/tex?x=H%27&#038;fg=00000&#038;s=0\" title=\"H'\" style=\"vertical-align:-20%;\" class=\"tex\" alt=\"H'\" \/><\/span>. Im linken Teil kann man nun den Strahlensatz anwenden und erh\u00e4lt:<br \/>\n<span class=\"tex\"><img decoding=\"async\" src=\"\/tex?x=%5Cfrac%7BBG%7D%7BC%27H%27%7D%20%3D%20%5Cfrac%7BAG%7D%7BD%27H%27%7D&#038;fg=00000&#038;s=0\" title=\"\\frac{BG}{C'H'} = \\frac{AG}{D'H'}\" style=\"vertical-align:-20%;\" class=\"tex\" alt=\"\\frac{BG}{C'H'} = \\frac{AG}{D'H'}\" \/><\/span> mit <span class=\"tex\"><img decoding=\"async\" src=\"\/tex?x=%5Coverline%7BBG%7D%3D600-h&#038;fg=00000&#038;s=0\" title=\"\\overline{BG}=600-h\" style=\"vertical-align:-20%;\" class=\"tex\" alt=\"\\overline{BG}=600-h\" \/><\/span>, <span class=\"tex\"><img decoding=\"async\" src=\"\/tex?x=%5Coverline%7BAG%7D%3Dh&#038;fg=00000&#038;s=0\" title=\"\\overline{AG}=h\" style=\"vertical-align:-20%;\" class=\"tex\" alt=\"\\overline{AG}=h\" \/><\/span>, <span class=\"tex\"><img decoding=\"async\" src=\"\/tex?x=%5Coverline%7BC%27H%27%7D%3Dh&#038;fg=00000&#038;s=0\" title=\"\\overline{C'H'}=h\" style=\"vertical-align:-20%;\" class=\"tex\" alt=\"\\overline{C'H'}=h\" \/><\/span> und <span class=\"tex\"><img decoding=\"async\" src=\"\/tex?x=%5Coverline%7BD%27H%27%7D%3D400-h&#038;fg=00000&#038;s=0\" title=\"\\overline{D'H'}=400-h\" style=\"vertical-align:-20%;\" class=\"tex\" alt=\"\\overline{D'H'}=400-h\" \/><\/span> ergibt sich <span class=\"tex\"><img decoding=\"async\" src=\"\/tex?x=%5Cfrac%7B600-h%7D%7Bh%7D%3D%5Cfrac%7Bh%7D%7B400-h%7D&#038;fg=00000&#038;s=0\" title=\"\\frac{600-h}{h}=\\frac{h}{400-h}\" style=\"vertical-align:-20%;\" class=\"tex\" alt=\"\\frac{600-h}{h}=\\frac{h}{400-h}\" \/><\/span>. Umgeformt und aufgel\u00f6st ergibt sich <span class=\"tex\"><img decoding=\"async\" src=\"\/tex?x=h%3D240&#038;fg=00000&#038;s=0\" title=\"h=240\" style=\"vertical-align:-20%;\" class=\"tex\" alt=\"h=240\" \/><\/span>.<\/p>\n<p>Die beiden Seilbahnen treffen sich also in 240 Metern H\u00f6he.<\/p>\n<p><\/div>\n<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Nach einiger Zeit hier mal wieder ein R\u00e4tsel, auch aus Professor Stewart&#8217;s Cabinet of Mathematical Curiosities S.124: In einem Bergtal in dem rechts und links steile Klippen aufragen gibt es zwei Seilbahnen, eine geht vom linken Talpunkt A zur rechten<span class=\"ellipsis\">&hellip;<\/span><\/p>\n<div class=\"read-more\"><a  href=\"https:\/\/www.fiveoclock.de\/?p=585\">Read more <span class=\"screen-reader-text\">Seilbahnen<\/span><span class=\"meta-nav\"> &#8250;<\/span><\/a><\/div>\n<p><!-- end of .read-more --><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[3],"tags":[],"_links":{"self":[{"href":"https:\/\/www.fiveoclock.de\/index.php?rest_route=\/wp\/v2\/posts\/585"}],"collection":[{"href":"https:\/\/www.fiveoclock.de\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.fiveoclock.de\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.fiveoclock.de\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.fiveoclock.de\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=585"}],"version-history":[{"count":1,"href":"https:\/\/www.fiveoclock.de\/index.php?rest_route=\/wp\/v2\/posts\/585\/revisions"}],"predecessor-version":[{"id":1020,"href":"https:\/\/www.fiveoclock.de\/index.php?rest_route=\/wp\/v2\/posts\/585\/revisions\/1020"}],"wp:attachment":[{"href":"https:\/\/www.fiveoclock.de\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=585"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.fiveoclock.de\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=585"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.fiveoclock.de\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=585"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}