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

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