Answer:
7031.25 m²
Step-by-step explanation:
The area of one pen will be largest when the area of a ractangle is the largest.
x - one side of rectangular area (x>0)
y - second side of rectangular area (y>0)
The farmer has 750 m of fencing, so if we subtract the sides x and fencing used for dividing into pens we receive:
y = 750 - 5x (750-5x>0 ⇒ x<150)
Area of the rectangle: A = x•y
A(x) = x•(750 - 5x) D=(0, 150)
A(x) = -5x² + 750x
A'(x) = -5•2x + 750•1 = -10x + 750 = -10(x - 75)
if x=75 then A'(x) = 0
so x = 75 is critical point
First Derivative Test:
A'(x) > 0 ⇔ -10(x-75)>0 ⇔ x-75<0 ⇔ x<75
A'(x) < 0 ⇔ -10(x-75)<0 ⇔ x-75>0 ⇔ x>75
A′(x)>0 to the left of x=75 and A′(x)>0 to the right of x=75 then x=75 is a maximum. {rational but also global in the interval (0,150)}
A(75) = 75•(750 - 5•75) = 75•375 = 28125 m²
So the largest possible area of each of the four pens is:
8125 : 4 = 7031.25 m²
