Answer to Question #655 in Calculus for Calculus

Question

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.

Answer 1

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) -> 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"