Use the sine ratio for a right triangle: $\sin (\theta )=\frac{\text{opposite}}{\text{hypotenuse}}$. Since $\theta =30^{\circ }$ and the hypotenuse is 10,

Because $\sin (30^{\circ })=\frac{1}{2}$,

Multiply both sides by 10:

So the correct answer is 5.

The ladder, wall, and ground form a right triangle. Since the ladder makes a $37^{\circ }$ angle with the ground, the ladder is the hypotenuse and the height up the wall is the side opposite the $37^{\circ }$ angle. Using sine,

Multiplying both sides by 10 gives

Therefore, the statement that must be true is choice A.

The height of the tree and the shadow form a right triangle. From the tip of the shadow, the angle of elevation to the top of the tree is $45^{\circ }$. In this triangle, the height of the tree is opposite the angle, and the shadow length of 24 feet is adjacent to the angle. Use tangent: $\tan (45^{\circ })=\frac{\text{opposite}}{\text{adjacent}}=\frac{h}{24}$. Since $\tan (45^{\circ })=1$, we get $1=\frac{h}{24}$, so $h=24$. The tree is 24 feet tall.

The 200 feet is the distance along the trail, so it represents the hypotenuse of a right triangle. The vertical elevation gain is the side opposite the $18^{\circ }$ angle. Using sine,

So,

Since $\sin (18^{\circ })\approx 0.309$,

This is about 62 feet, so the correct answer is $62$.

Since angle $Q$ is $90^{\circ }$, the legs of the triangle are $PQ$ and $QR$. Relative to angle $P$, the opposite side is $QR$ and the adjacent side is $PQ$. Therefore,

Substitute the given values:

Cross-multiply:

Then divide by 5:

So the correct answer is $36$.