A rancher has 400 feet of fencing with which to enclose two adjacent rectangular corrals....

To get the dimensions of our rectangle, we must first determine x and y.
We can see from the sketch that we have
${\displaystyle 2x+3y=400\Rightarrow x=200-\frac{3}{2}y}$
Our area is
${\displaystyle A(x,y)=2xy}$
As we can see from the first equation, our function is
${\displaystyle A(x,y)=2xy}$
We can see from the first equation that we have So our function is
${\displaystyle A(x,y)=2xy}$
${\displaystyle A\left(y\right)=2\cdot (200-\frac{3}{2}y)\cdot y}$
${\displaystyle A\left(y\right)=(400-3y)y}$
${\displaystyle \Rightarrow A\left(y\right)=-3y^2+400y}$
The first step is to identify critical numbers and solve them -
${\displaystyle A^{\prime }\left(y\right)=0}$
This leads to 
${\displaystyle A^{\prime }\left(y\right)=-6y+400}$
So our equation becomes
${\displaystyle -6y+400=0\Rightarrow y=\frac{200}{3}}$
When we plug this into expression for x, we get
${\displaystyle x=200-\frac{3}{2}y\Rightarrow x=100}$
As a result, our ultimate dimensions are
${\displaystyle x=100;y=\frac{200}{3}}$