jmc

Let the width of the rectangle be $w$. Then the length of the rectangle is $2w$. We can apply the Pythagorean Theorem to two sides of this rectangle and get that the length of the diagonal is $5\sqrt{5}=\sqrt{w^2+(2w{)}^2}=\sqrt{5w^2}$. We therefore have $5\sqrt{5}=w\sqrt{5}$, which means that $w=5$. This means that the length of the rectangle is 10, so its area is $5\cdot 10=\boxed{50}$.