There is a triangular park with vertices at $O$, $A$, $B$. At vertex $O$ there is an observation tower 150 meters high. A person standing on the observation tower observes the other two vertices $A$, $B$ on the ground and the midpoint $C$ of $\overline{AB}$, measuring angles of depression of $30^{\circ}$, $60^{\circ}$, $45^{\circ}$ respectively. The area of this triangular park is （15）（16）（17）（18）$\sqrt{(19)}$ square meters. (Express as a simplified radical)

China's Tiger Hill Tower, Pearl Tower, and Italy's Leaning Tower of Pisa are three famous leaning towers with tower heights of 48, 19, and 57 meters respectively, and offset distances of 2.3, 2.3, and 4 meters respectively. Their tilt angles are denoted as $\theta_1{}^{\circ}, \theta_2{}^{\circ}$, and $\theta_3{}^{\circ}$ respectively. Compare the size relationship of $\theta_1, \theta_2$, and $\theta_3$. (Non-multiple choice, 4 points)
Note: The tilt angle $\theta^{\circ}$ is the angle between the tower body and a vertical dashed line ($0 \leq \theta < 90$), and the offset distance is the distance from the tower top to the vertical dashed line.

Suppose there are two iron towers with equal tower heights. Their tilt angles $\alpha^{\circ}, \beta^{\circ}$ satisfy $\sin\alpha^{\circ} = \frac{1}{5}$ and $\sin\beta^{\circ} = \frac{7}{25}$ respectively. It is known that the offset distances of the two towers differ by 20 meters. Find the difference in the distance from the tower tops to the ground. (Non-multiple choice, 6 points)
Note: The tilt angle $\theta^{\circ}$ is the angle between the tower body and a vertical dashed line ($0 \leq \theta < 90$), the offset distance is the distance from the tower top to the vertical dashed line, and the distance from the tower top to the ground is the vertical height.

There are two tall buildings on the ground, Building A and Building B. It is known that Building A is taller than Building B, and the horizontal distance between the two buildings is 150 meters. A person pulls a rope from the top of Building A to the top of Building B, and measures the angle of depression to the top of Building B from the top of Building A as $22^{\circ}$. Assuming the rope is pulled straight, which of the following options is closest to the length of the rope (unit: meters)? (Note: The angle of depression is the angle between the line of sight and the horizontal line when looking down at an object)
(1) $150$ (2) $150 \sin 22^{\circ}$ (3) $150 \cos 22^{\circ}$ (4) $\frac{150}{\cos 22^{\circ}}$ (5) $\frac{150}{\sin 22^{\circ}}$

As shown in the diagram on the right, there is a $\triangle ABC$. It is known that the altitude $\overline{AD} = 12$ on side $\overline{BC}$, and $\tan \angle B = \frac{3}{2}$, $\tan \angle C = \frac{2}{3}$. What is the length of $\overline{BC}$?
(1) 20
(2) 21
(3) 24
(4) 25
(5) 26

Captain Temel will take the tourists on his boat from island A to island B in the morning, from island B to island C at noon, and from island C to island A in the evening.
The points where the boat will dock at the islands are marked as the vertices of a triangle ABC where side AB equals side BC, as shown in the figure.
Since Captain Temel knows he will travel in the dark on the return journey, as he travels from A to B and from B to C, he notes on a piece of paper the angle between the compass needle pointing north and the path he follows.
Accordingly, how should Captain Temel set his compass to go from C to A?

A seesaw on a flat ground as shown in Figure 1 consists of a straight segment 30 units long and a straight support 9 units long located at the exact center of this segment.
As shown in Figure 2, when the left end of the seesaw touches the ground, a shaded region in the shape of a right trapezoid is formed on the right side.
Accordingly, what is the perimeter of this trapezoid in units?
A) 54
B) 55
C) 56
D) 57
E) 58