Question

A rancher has 264 feet of fencing to enclose two adjacent rectangular corrals. She will form the corrals by building one large rectangle with the fencing, and then dividing it down the middle with the fencing. What dimensions of the large rectangle will produce the largest total area?

Answer 1

Answer:

Width $= 5.07$ feet

Length $= 125.40$ feet

Explanation:

Let the length of two small rectangles be "y" and width be "X"

There are total 4 y and 3 X  to cover with fence.

Hence,  perimeter of area to be covered by fence is equal to $264$

Thus,

$4Y + 3X = 264$

Diving the entire equation by $4$,  we get -

$Y + \frac{3}{4} X = 66\\Y + 0.75 X = 66$

Area of the rectangle is equal to $XY$

$XY = A$

$X (66-0.65X) = A$

Differentiating both sides with respect to X we get -

$\frac{dA}{dx} = 0\\\frac{d(66X - 0.65X^2)}{dX} = 0\\66 - 1.3 X = 0\\X = 5.07$

Therefore ,

$Y = 66 -0.65 X\\Y = 66 - 0.65 * 5.07\\Y = 62.70$

Therefore ,

Length of large rectangle is equal to $2 * 62.70 = 125.4$ feet

and width of large rectangle is equal to $5.07$ feet