A rectangular field on a farm is to be fenced in using the wall of the barn for one side and 200 meters of fencing for the other three sides. The area A (x) in square meters of the field is a function of the length x in meters of each of the sides perpendicular to the barn wall

Rectangular area as a function of x : A (x) = 200*x + 2*x²

A (max) = 5000 m²

Dimensions:

x = 50 m

l = 100 m

Step-by-step explanation:

"x" is the length of the perpendicular side to the wall of the rectangular area to be fenced, and we call "l" the other side (parallel to the wall of the barn) then:

A (r) = x * l and the perimeter of the rectangular shape is

P = 2*x + 2*l but we won't use any fencing material along the wll of the barn therefore

P = 2*x + l ⇒ 200 = 2*x + l ⇒ l = 200 - 2*x (1)

And the rectangular area as a function of x is:

A (x) = x * (200 - 2*x) ⇒ A (x) = 200*x + 2*x²

Taking derivatives on both sides of the equation we get:

A' (x) = 200 - 4*x ⇒ A' = 0

Then 200 - 4*x = 0 ⇒ 4*x = 200 ⇒ x = 50 m

We find the l value, plugging the value of x in equation (1)

l = 200 - 2*x ⇒ l = 200 - 2*50 ⇒ l = 100 m

A (max) = 100*50

A (max) = 5000 m²