If the farmer has 250 feet of fencing to create a rectangular pen, define a function f that expresses the area of the field (measured in square feet) as a function of the width of the side of the field w (measured in feet). What is the maximum area possible?

Maximum area possible

f (max) = 3906,25 ft²

Dimensions:

a = 62,5 ft

w = 62,5 ft

Step-by-step explanation:

Perimeter of the rectangular fencing P = 250 feet

And sides of the rectangle a and w (width of rectangle)

Then

A = a*w

2a + 2w = 250 ⇒ a = (250 - 2w) / 2 ⇒ a = 125 - w

f (w) = (125 - w) * w f (w) = 125w - w²

Taking derivatives both sides of the equation

f' (w) = 125 - 2w f' (w) = 0 125 - 2w = 0

w = 125/2

w = 62,5 ft ⇒ a = 125 - 62,5

a = 62,5 ft

f (max) = (62,5) ²

f (max) = 3906,25 ft²