Answer
$37\;\rm km$
Work Step by Step
We need to draw the force diagram of the airplane, as shown in the figures below.
Now we need to apply Newton's second law to find the radius of the path.
$$\sum F_x=F_L\sin\theta=ma_r=m\dfrac{v^2}{R}$$
Thus,
$$F_L\sin\theta= m\dfrac{v^2}{R}\tag 1$$
$$\sum F_y=F_L\cos\theta-mg=ma_y=m(0)=0$$
Thus,
$$F_L=\dfrac{mg}{\cos\theta}$$
Plugging into (1);
$$\dfrac{\color{red}{\bf\not} mg}{\cos\theta}\sin\theta= \color{red}{\bf\not} m\dfrac{v^2}{R} $$
$$ g \tan\theta= \dfrac{v^2}{R} $$
Therefore,
$$R= \dfrac{v^2}{g \tan\theta}$$
Noting that the diameter of this circular path is double its radius. So that
$$D=2R= \dfrac{2v^2}{g \tan\theta}$$
Plugging the known but do not forget to convert them to SI units.
$$D= \dfrac{2 \times 178.89^2}{9.8 \tan10^\circ}$$
$$D=37.04\times10^3\;\rm m\approx\color{red}{\bf37}\;\rm km$$
