Question

A rancher with 750 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle.

(a) Find a function that models the total area of the four pens.
(b) Find the largest possible total area of the four pens.

Answer 1

a) A(r) = ( 1/2) * (750*x - 5*x²)

b) Dimensions

x = 75 ft

y = 187,5 ft

A (max) = 14062,5 ft²

Explanation:

Fencing material available 750 ft

Rectangula area A(r)

Let x and y dimensions of rectangular area, and x the small side of the rectangle, then

The perimeter of the rectangle is

P(r) = 2*x + 2*y (1)

To get the four pens we have to place three more fence in between the two x sides of the rectangle, in such way that the total fence is

P(r) + 3*x = 750

So 2*x + 2*y + 3*x = 750 ⇒ 5*x + 2*y = 750 ⇒ y = (750- 5x)/2

Plugging that value in (1)

A(r) = x* y

A(r) = x* ( 750 - 5*x)/ 2 ⇒ A(r) = ( 1/2) * (750*x - 5*x²)

Taking derivatives in both sides of the equation we get:

A´(r) = (1/2) * ( 750 - 5*x) ⇒ A´(r) = 0 ⇒ (1/2) * ( 750 - 10*x) = 0

750 - 10*x = 0 ⇒ x = 750/10 ⇒ x = 75 ft

And y would be

y = (750- 5x)/2 ⇒ y = (750 - 5*75) / 2 ⇒ y = 375 / 2

y = 187,5 ft

A(max) = 187,5*75

A(max) = 14062,5 ft²