Please Help with Optimization Problem

Question

A rancher has 1000m of fencing to enclose two rectangular corrals. The corrals have the same dimensions and one side in common. What dimensions will maximize the enclosed area? Show all work.

Answer 1

4x+3y = 1000

3y =1000-4x

y= (1000-4x)/3

Area = 2xy = 2x(1000-4x)/3 = 2000x/3 -8x^2/3

take the derivative and set = 0

2000/3-16x/3 = 0 multiply by 3

16x = 2000

x = 2000/16 = 1000/8 = 125 feet

y = (1000-4x)/3 = (500)/3 = 166 2/3 feet

max area = 2xy = 2(125)(166 2/3) = 41,666 2/3 square feet