Question

suppose a farmer encloses a rectangular region of a land next to a river. fencing will be used on 3 sieds, and none is needed along the river. tje farmer ha 180 feet of fencing available to use. using x and y for the dimensions of the rectangle. the equation for the amount of fencing used on each side is 2x u

Answer 1

Dimensions :

x (the longer side, only one side with fence ) = 90 ft

y ( the shorter side two sides with fence ) = 45 ft

Total fence used 45 * 2 + 90 = 180 ft

A(max) =

Step-by-step explanation: If a farmer has 180 ft of fencing to encloses a rectangular area with fence in three sides and the river on one side, the farmer surely wants to have a maximum enclosed area.

Lets call "x" one the longer side ( only one of the longer side of the rectangle will have fence, the other will be along the river and won´t need fence. "y" will be the shorter side

Then we have:

P = perimeter = 180 = 2y + x ⇒ y = ( 180 - x ) / 2 (1)

And A (r) = x * y

A(x) = x * ( 180 - x ) /2 ⇒ A(x) = (180/2) *x - x² / 2

Taking derivatives on both sides of the equation :

A´(x) = 90 - x

Then if A´(x) = 0 ⇒ 90 - x = 0 ⇒ x = 90 ft

and from : y = ( 180 - x ) / 2 ⇒ y = 90/2

y = 45 ft

And

A(max) = 90 * 45 = 4050 ft²