24.5 The slant height of a cone is $l$, and the apex angle of the axial section is $120^{\circ}$. Then the range of the area of the cone's section passing through the apex is
(A) $\left(0, \frac{\sqrt{3}}{4} l^{2}\right]$.
(B) $\left[\frac{\sqrt{3}}{4} l^{2}, \frac{l^{2}}{2}\right]$.
(C) $\left(0, \quad \frac{l^{2}}{2}\right]$.
(D) $\left[0, \frac{l^{2}}{2}\right]$.
（2nd "Hope Cup" National Mathematics Invitational Competition, 1991）
［Solution］As shown in the figure, the cross-section of the cone through the apex is an isosceles triangle with legs of length $l$. Let the vertex angle be $\theta$, then its area is
$$
S=\frac{1}{2} l^{2} \sin \theta,
$$

The vertex angle of the axial section is $120^{\circ}$, so $0^{\circ}<\theta \leqslant 120^{\circ}$.
Therefore, $0<S \leqslant \frac{1}{2} l^{2} \sin 90^{\circ}=\frac{1}{2} l^{2}$.
Hence, the correct choice is (C).