Answer to Question #655 in Calculus for Calculus
The enclosed is to equal 1800 feet.
Find the minimum perimeter and the dimensions of the corresponding enclosure.
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) -> min
Since y = 1800/x (from the equation of the area), P = 2*(x + 1800/x) -> min
The derivative: P'=2*(1-1800/x^2)=0 => x^2=1800 => x=30*sqrt(2) => y=1800/x=30*sqrt(2) => 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.
* "sqrt" means "square root"
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