# 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.

The enclosed is to equal 1800 feet.

Find the minimum perimeter and the dimensions of the corresponding enclosure.

Expert's answer

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"

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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