Answer
$$x=25,\ \ y=25$$
Work Step by Step
Let $x$ be the length and width be $y $ then the perimeter given by
$$P=2(x+y)\ \ \Rightarrow \ 50=x+y $$
Since are give by
$$A=xy=x(50-x)=50x-x^2$$
The goal is to maximize $A$ , since
$$ A'(x)=50-2x$$
Then $A'(x)=0$ for $x=25$ , since $A'(x)>0$ for $x<25$ , $A'(x)<0$ for $x>25$, hence $A(x)$ has maximum at $x=25$ and $y=50-x=25$