a rancher has 1200 feet of fencing to enclose two adjacent rectangular corrals of equal lengths and widths…

Explanation:

Step1: Define variables

Let the length of each corral be (x) and the width be (y). The total fencing used is (3x + 4y=1200), so (y = 300-\frac{3}{4}x). The area (A=2xy).

Step2: Substitute (y) into area formula

Substitute (y = 300-\frac{3}{4}x) into (A), we get (A=2x\left(300-\frac{3}{4}x\right)=600x-\frac{3}{2}x^{2}).

Step3: Find the derivative of (A)

Differentiate (A) with respect to (x), (A^\prime=600 - 3x).

Step4: Set derivative equal to zero

Set (A^\prime = 0), (600-3x=0), then (x = 200).

Step5: Find (y)

Substitute (x = 200) into (y = 300-\frac{3}{4}x), (y=300-\frac{3}{4}\times200 = 150).

Step6: Calculate the maximum area

The maximum area (A = 2xy=2\times200\times150 = 60000) square feet.

Answer:

(60000) square feet