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# Answer to Question #655 in Calculus for Calculus

Question #655
A three-sided fence is to be built next to a straight section of river,witch forms the fourth side of a rectangular of a region. The enclosed is to equal 1800 feet. Find the minimum perimeter and the dimensions of the corresponding enclosure.
1
2010-09-21T06:33:30-0400
Let x and y be the sides of the rectangular region.
The enclosed area is to equal 1800 square feet, thus: x*y = 1800.
The perimeter of the rectangular region is: P = 2*(x+y). And the task is to minimize this function.
Thus, P = 2*(x+y) -&gt; min
Since y = 1800/x (from the equation of the area), P = 2*(x + 1800/x) -&gt; min
The derivative: P&#039;=2*(1-1800/x^2)=0 =&gt; x^2=1800 =&gt; x=30*sqrt(2) =&gt; y=1800/x=30*sqrt(2) =&gt; P=2*(x+y)=120*sqrt(2)

Answer: the enclosure must be a square with the side 30*sqrt(2) feet and the perimeter 120*sqrt(2) feet.

* &quot;sqrt&quot; means &quot;square root&quot;

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