https://artofproblemsolving.com/wiki/index.php?title=2005_PMWC_Problems/Problem_I13&feed=atom&action=history 2005 PMWC Problems/Problem I13 - Revision history 2021-10-18T15:08:49Z Revision history for this page on the wiki MediaWiki 1.31.1 https://artofproblemsolving.com/wiki/index.php?title=2005_PMWC_Problems/Problem_I13&diff=17361&oldid=prev Azjps: sol 2007-10-02T21:06:51Z <p>sol</p> <p><b>New page</b></p><div>== Problem ==<br /> Sixty meters of rope is used to make three sides of a rectangular camping area with a long wall used as the other side. The length of each side of the [[rectangle]] is a [[natural number]]. What is the [[maximum|largest]] area that can be enclosed by the rope and the wall? <br /> <br /> == Solution ==<br /> Let &lt;math&gt;l&lt;/math&gt; be the side [[parallel]] to the wall, and &lt;math&gt;w&lt;/math&gt; adjacent to the wall. Then &lt;math&gt;l + 2w = 60 \Longrightarrow l = 60 - 2w&lt;/math&gt;. The area of the rectangle is &lt;math&gt;lw = -2w^2 + 60w&lt;/math&gt;; this is a [[quadratic equation]], which we can maximize using the formula &lt;math&gt;\frac{-b}{2a} = 15&lt;/math&gt;. Hence the area is &lt;math&gt;-2(15)^2 + 60(15) = 450&lt;/math&gt;.<br /> <br /> == See also ==<br /> {{PMWC box|year=2005|num-b=I12|num-a=I14}}<br /> <br /> [[Category:Introductory Algebra Problems]]</div> Azjps