What dimensions of the large rectangle will produce the largest total area? What is the maximum total area?

Question

A rancher has 360 feet of fencing to enclose two adjacent rectangular corrals. She will form the corrals by building one large rectangle with the fencing, and then diving it down the middle with the fencing.

Answer 1

3x+2y=360 ft

y = (360-3x)/2

Area = xy = x(180-3x/2= 180x -3x^2/2

A' =180-3x = 0

x = 180/3 = 60 feet for 3 sides

y =180-3x/2 = 180-90 =90 feet for 2 sides

Answer 2

3L + 2W = 360 ft

A = L×W and W = ½(360 – 3L)

which means that

A = ½L(360 – 3L) = 180L – (3/2)L²

which describes a parabola

A = aL² + bL

that opens downward,

because it's leading coefficient (a) is negative.

The parabola reaches a maximum at its vertex point

where L = –b/2a

So maximum area will occur at

ft

W = ½(360 – 3L) = ½(360 – 3(60)) = ½(180) = 90 ft

Answer

The dimensions of the rectangle producing the largest total area are

L = 60 ft

W = 90 ft

Answer 3

Let a be the length of the total rectangular area, and let b be the width. Dividing it in the middle makes each corral be a/2 by b.

The total fencing is 2a + 3b = 360. This is because the perimeter of the large rectangle is 2a + 2b, and the division down the middle has length b.

Solve for b in terms of a to get b = 120 - (2/3)a.

Meanwhile, the total area is ab. Plugging in our expression for b in terms of a, we get that the area is a(120 - (2/3)a). Multiplying out, we get 120a - (2/3)a².

To find the maximum area, we take the derivative of this last expression and set it equal to zero: 120 - (4/3)a = 0.

Solving for a, we get a=90.

Plugging back into the expression for b in terms of a, we get b=60.

So the dimensions of the large rectangle are 90 by 60, and the maximal total area is 90x60 = 5400. Hope this helps!