{"id":726,"date":"2013-10-28T14:13:46","date_gmt":"2013-10-28T06:13:46","guid":{"rendered":"http:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/?p=726"},"modified":"2024-02-16T21:12:10","modified_gmt":"2024-02-16T13:12:10","slug":"more-on-the-rectangle-puzzle","status":"publish","type":"post","link":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/2013\/10\/more-on-the-rectangle-puzzle\/","title":{"rendered":"More On The Rectangle Puzzle"},"content":{"rendered":"<p>Last week, I posted a <a title=\"Rectangles and Right Triangles\" href=\"http:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/2013\/10\/rectangles-and-right-triangles\/\">solution to a puzzle<\/a> &#8211; how can you find a rectangle whose area equals its perimeter?<\/p>\n<p>This week, I&#8217;ll post a simpler solution.<\/p>\n<p><!--more--><\/p>\n<p>Again, we need a bit of algebra, but the algebra will be simpler this time. I&#8217;ll call the sides of the rectangle A and B, but instead of inventing a new variable S to equal the area and perimeter, I&#8217;ll make T equal to the ratio A\/B.<\/p>\n<p>Now, this is a bit nonintuitive. After all, this puzzle doesn&#8217;t seem to have anything to do with the ratio of the sides. However, let&#8217;s carry on.<\/p>\n<ul>\n<li>If T is A\/B, then A is BT.<\/li>\n<li>The area AB, that is, B<sup>2<\/sup>T.<\/li>\n<li>The perimeter is 2A + 2B, that is, 2BT+2B, that is, B times 2T+2.<\/li>\n<li>Since the area equals the perimeter, this means B<sup>2<\/sup>T equals B(2T+2).<\/li>\n<li>If I cancel out B, I get BT = 2T+2<\/li>\n<\/ul>\n<p>But BT is A, so I&#8217;ve instantly got a formula for A. Specifically, A = 2T + 2. Dividing this by T gives me B = 2 + 2\/T.<\/p>\n<p>So, I have another puzzle solution factory &#8211; much simpler, you&#8217;ll agree, than last week&#8217;s. The puzzle solution factory works like this:<\/p>\n<ul>\n<li>Choose a value of T,<\/li>\n<li>Work out 2 + 2T and 2 + 2\/T<\/li>\n<li>Those are the sides of the rectangle!<\/li>\n<\/ul>\n<p>Let&#8217;s try it.<\/p>\n<ul>\n<li>Choose T=1<\/li>\n<li>2+2T = 4 and 2+2\/T is also 4.<\/li>\n<li>So the 4&#215;4 square has perimeter equal to its area!<\/li>\n<\/ul>\n<p>And again&#8230;<\/p>\n<ul>\n<li>Choose T=4<\/li>\n<li>2+2T is 10, and 2+2\/T is 2.5.<\/li>\n<li>So, a 2.5 x 10 rectangle has area equal to its perimeter!<\/li>\n<\/ul>\n<p>An obvious moral of this little story is that there&#8217;s often more than one way to solve a math problem, and some ways are harder than others. But we kind of knew that already. Next week, I&#8217;ll talk about a much more powerful thing this little puzzle teaches us about math.<\/p>\n<p>Stay tuned!<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Last week, I posted a solution to a puzzle &#8211; how can you find a rectangle whose area equals its perimeter? This week, I&#8217;ll post a simpler solution.<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[7,35,47],"_links":{"self":[{"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/posts\/726"}],"collection":[{"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/comments?post=726"}],"version-history":[{"count":3,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/posts\/726\/revisions"}],"predecessor-version":[{"id":729,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/posts\/726\/revisions\/729"}],"wp:attachment":[{"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/media?parent=726"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/categories?post=726"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/tags?post=726"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}