$\sqrt{(224)_r} = (13)_r$
Converting $r$ base to decimal
$\sqrt{2\times r^2 + 2\times r + 4} = 1 \times r + 3$
Take square on both sides
$2r^2 + 2r + 4 = r^2 + 6r+ 9$
$\implies r^2 - 4r -5 = 0$
$\implies r^2 -5r +r - 5 = 0$
$\implies (r-5)(r+1)=0$
$r$ being a base, it can not be $-1.$
So, $C.$ $r = 5$ is correct answer