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Let's go through each of these trigonometric problems step-by-step:

### Problem 3
Matt is on top of a cliff that is 305 feet above a lake. The angle of depression to a boat on the lake is 42°. How far is the boat from Matt?

1. **Set up the triangle:** The angle of depression from Matt to the boat is the same as the angle of elevation from the boat to Matt. This is a right triangle where:
   - Opposite side = 305 feet (height of the cliff).
   - Adjacent side = the distance from Matt to the boat.

2. **Use the tangent function:** 
   \[
   \tan(42^\circ) = \frac{305}{\text{distance}}
   \]

3. **Solve for distance:**
   \[
   \text{distance} = \frac{305}{\tan(42^\circ)}
   \]
   Calculate this value to get the distance.

### Problem 4
A ladder is 20 ft. long and leaning against a building. The angle between the ladder and the ground is 75°. How far is the bottom of the ladder from the base of the building?

1. **Set up the triangle:** 
   - Hypotenuse = 20 ft (length of the ladder).
   - Adjacent side = distance from the bottom of the ladder to the base of the building.

2. **Use the cosine function:**
   \[
   \cos(75^\circ) = \frac{\text{adjacent}}{20}
   \]

3. **Solve for the adjacent side (distance from base of building):**
   \[
   \text{distance} = 20 \cdot \cos(75^\circ)
   \]
   Calculate this value to find the distance.

### Problem 5
A ladder 30 ft long needs to reach 27 ft up a building. What should the angle off the vertical be?

1. **Set up the triangle:**
   - Hypotenuse = 30 ft.
   - Opposite side = 27 ft (height the ladder needs to reach).

2. **Use the sine function:**
   \[
   \sin(\theta) = \frac{27}{30}
   \]

3. **Solve for \(\theta\):**
   \[
   \theta = \arcsin\left(\frac{27}{30}\right)
   \]
   This gives the angle off the horizontal. Subtract from 90° to get the angle off the vertical.

### Problem 6
You are standing 50 meters from a hot air balloon that is preparing to take off. The angle of elevation to the top of the balloon is 28°. Find the height of the balloon.

1. **Set up the triangle:** 
   - Adjacent side = 50 meters (distance from you to the balloon).
   - Opposite side = height of the balloon.

2. **Use the tangent function:**
   \[
   \tan(28^\circ) = \frac{\text{height}}{50}
   \]

3. **Solve for height:**
   \[
   \text{height} = 50 \cdot \tan(28^\circ)
   \]
   Calculate this value to get the height of the balloon.

Would you like detailed calculations for each of these solutions?

### Follow-up Questions
1. How would these calculations change if the angles were increased by 10 degrees?
2. What trigonometric functions are best suited for right triangles?
3. How can angle of depression be used in real-life applications beyond navigation?
4. How would you calculate distances if you only had the hypotenuse and one angle?
5. What would happen to the solutions if the height of the cliff or ladder length changed?

### Tip
For trigonometric problems, remember to use the correct function based on which sides of the triangle you know: sine for opposite/hypotenuse, cosine for adjacent/hypotenuse, and tangent for opposite/adjacent.

Matt is standing on top of a cliff 305 feet above a lake. The measurement of the angle of depression to a boat on the lake is 42°. How far is the boat from Matt? A ladder that is 20 ft. long is leaning against the side of a building. If the angle formed between the ladder and the ground is 75°, how far is the bottom of the ladder from the base of the building? A ladder that is 30 ft long needs to reach 27 ft up a building. What should the angle off of the vertical be? You are standing 50 meters from a hot air balloon that is preparing to take off. The angle of elevation to the top of the balloon is 28°. Find the height of the balloon.

Explore trigonometry with real-life scenarios involving angles of depression and elevation. Problems include calculating distances and heights using trigonometric ratios with angles, ladders, and balloon heights. Suitable for students in Grades 9-11.

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