Solve the following trigonometry problems: 1. A 13m ladder leans against a wall, its base is 5m from the wall. How high up the wall does it reach? 2. From a cliff 15m high, the angle of depression to

The ladder, the wall, and the ground form a right-angled triangle.

• Length of the ladder (hypotenuse) = 13 m
• Distance from the base of the wall to the foot of the ladder (adjacent side) = 5 m
• Height the ladder reaches up the wall (opposite side) = $h$ (unknown)

The Pythagorean theorem states $a^2 + b^2 = c^2$, where $c$ is the hypotenuse and $a$ and $b$ are the other two sides.

The ladder reaches $\boxed{\text{12 m}}$ up the wall.

: Angle of depression of a boat

The cliff, the sea level, and the line of sight to the boat form a right-angled triangle. The angle of depression from the top of the cliff to the boat is equal to the angle of elevation from the boat to the top of the cliff.

• Height of the cliff (opposite side to the angle of elevation) = 15 m
• Angle of depression = $35^\circ$ (which is also the angle of elevation from the boat)
• Distance from the base of the cliff to the boat (adjacent side) = $x$ (unknown)

The tangent of an angle in a right-angled triangle is the ratio of the opposite side to the adjacent side:

We are given $\tan 35^\circ = 0.7$.

The distance from the base of the cliff to the boat is $\boxed{\text{21.43 m}}$.

: Angle of depression to a swimmer

The cliff, the sea level, and the line of sight to the swimmer form a right-angled triangle.

• Height of the cliff (opposite side) = 30 m
• Horizontal distance to the swimmer (adjacent side) = 15 m
• Angle of depression = $\theta$ (unknown)

We are given:
$\tan 64^\circ = 2.0503$
$\tan 65^\circ = 2.1444$
$\tan 66^\circ = 2.2460$
Since $\tan(\theta) = 2$, the angle $\theta$ is between $63^\circ$ and $64^\circ$. Based on the provided options, the closest integer angle would be $63^\circ$ if we were to round, but since 2 is not exactly one of the given values, we can state the exact value of $\theta$.

Using a calculator, $\theta \approx 63.43^\circ$.
However, if we are to choose from the given values, none of them exactly match $\tan(\theta) = 2$. If the question implies finding the closest integer angle from the given table, then 2 is closest to 2.0503 (for $64^\circ$) than to 2.1444 (for $65^\circ$). But 2 is also less than 2.0503.
Let's assume the question expects us to find the angle whose tangent is exactly 2.
The angle of depression is $\boxed{\arctan(2) \approx 63.43^\circ}$.

: Height of a tree

We have a right-angled triangle formed by Bill's eye level, the horizontal distance to the tree, and the height from Bill's eye level to the top of the tree.

• Bill's height = 1.5 m
• Horizontal distance from Bill to the tree (adjacent side) = 5 m
• Angle of elevation = $20^\circ$
• Height from Bill's eye level to the top of the tree = $h'$ (unknown)

We are given $\tan 20^\circ = 0.36$.

$h' = 1.8$ m

The total height of the tree is Bill's height plus $h'$.
Total height = Bill's height + $h'$
Total height = $1.5 \text{ m} + 1.8 \text{ m}$
Total height = 3.3 m
The height of the tree is $\boxed{\text{3.3 m}}$.