15) If the lateral surface of a right cylinder is developed, a rectangle is obtained whose diagonals are of length $l$ and form an angle of $30^{\circ}$ with the base of the rectangle. The volume of the cylinder is
(A) $\frac{1}{16} \frac{l^{3}}{\pi}$
(B) $\frac{3}{8} l^{3}$
(C) $\frac{3}{4} l^{3}$
(D) $\frac{3}{32} \frac{l^{3}}{\pi}$

(E) the above answers are all incorrect.
15) The answer is (D).

![](https://cdn.mathpix.com/cropped/2024_04_17_dcf3bd4e9dbd26cb9fe5g-3.jpg?height=368&width=488&top_left_y=393&top_left_x=1435)

It is immediately clear that the height \( h \) of the rectangle, and therefore that of the cylinder, is \( \frac{l}{2} \) and that the base is \( l \frac{\sqrt{3}}{2} \). A circumference that is \( l \frac{\sqrt{3}}{2} \) long has a radius \( r = l \frac{\sqrt{3}}{4 \pi} \), so the volume of the cylinder is

\[
V = \pi r^{2} h = \pi l^{2} \frac{3}{16 \pi^{2}} \frac{l}{2} = \frac{3}{32} \frac{l^{3}}{\pi}
\]