Trigonometry problems involving angles of elevation and depression with diagrams.

Let’s solve each problem one by one. I’ll work through them step by step, check my math, and then give the final answer for each.

---

Problem 6:
> The angle of elevation from a boat to the top of a 50-meter cliff with a lighthouse on it is 27°. How far is the boat from the base of the cliff?

We’re told:
- Height = 50 meters (opposite side)
- Angle of elevation = 27°
- We need to find distance from boat to base of cliff → that’s the adjacent side.

Use tangent:
tan(θ) = opposite / adjacent
→ tan(27°) = 50 / x
→ x = 50 / tan(27°)

Calculate tan(27°): ≈ 0.5095
x = 50 / 0.5095 ≈ 98.13 meters

✔ Check: Makes sense — if angle is small, distance should be larger than height. 98 > 50 → good.

---

Problem 7:
> A green whale shark swims 22 feet below sea level. If the shadow is cast 35 feet from the surface, what is the angle of depression of its boat to the shark?

Wait — this wording is tricky. Let me re-read:

“A green whale shark swims 22 feet below sea level. If the shadow is cast 35 feet from the surface...”

Actually, looking at the diagram description (even though we don’t describe images), it seems like:

- Shark is 22 ft underwater (vertical leg)
- Horizontal distance from point above shark to where shadow hits surface? Or maybe the boat is 35 ft away horizontally?

But the question says: “what is the angle of depression of its boat to the shark?”

Assume:
- Boat is on surface.
- Shark is 22 ft directly below some point.
- But shadow is cast 35 ft from surface — probably meaning horizontal distance between boat and point above shark is 35 ft? That makes more sense.

So: right triangle with:
- Opposite = 22 ft (depth)
- Adjacent = 35 ft (horizontal distance)
- Angle of depression = angle from boat down to shark → same as angle of elevation from shark to boat.

So tan(θ) = opposite/adjacent = 22/35 ≈ 0.6286
θ = arctan(0.6286) ≈ 32.16°

✔ Check: Reasonable — not too steep, since 22 < 35.

---

Problem 8:
> A 15-foot cable supporting a circus tent is staked into the ground at an angle of elevation of 42°. How far from the tent must the stake be placed?

Cable = hypotenuse = 15 ft
Angle of elevation = 42° → angle between ground and cable
We want adjacent side (distance from tent to stake)

cos(θ) = adjacent / hypotenuse
→ cos(42°) = x / 15
→ x = 15 * cos(42°)

cos(42°) ≈ 0.7431
x ≈ 15 * 0.7431 ≈ 11.15 ft

✔ Check: Less than 15 → makes sense.

---

Problem 9:
> A bird is flying at a height of 34 feet and spots a 4-foot fence on which to perch. If the top of the fence is at a 28° angle of depression from the bird, how far must the bird fly before it can land?

Bird is at 34 ft high. Fence is 4 ft high → so vertical drop to top of fence = 34 - 4 = 30 ft

Angle of depression = 28° → this is the angle from bird’s line of sight down to top of fence.

In the right triangle:
- Opposite side = 30 ft (vertical drop)
- Angle = 28°
- We want adjacent side = horizontal distance bird must fly

tan(28°) = opposite / adjacent = 30 / x
→ x = 30 / tan(28°)

tan(28°) ≈ 0.5317
x ≈ 30 / 0.5317 ≈ 56.42 ft

✔ Check: Since angle is less than 45°, adjacent should be longer than opposite → 56 > 30 → correct.

---

Problem 10:
> A 19-meter ladder is inclined against a brick wall at an angle of 25°. If the top of the ladder reaches the top of the wall, how tall is the wall?

Ladder = hypotenuse = 19 m
Angle with ground = 25°
Wall height = opposite side

sin(25°) = opposite / hypotenuse = h / 19
→ h = 19 * sin(25°)

sin(25°) ≈ 0.4226
h ≈ 19 * 0.4226 ≈ 8.03 m

✔ Check: Wall shorter than ladder → yes, 8 < 19 → reasonable.

---

Now, let’s compile all answers clearly.

Final Answer:
6) 98.1 meters
7) 32.2 degrees
8) 11.1 feet
9) 56.4 feet
10) 8.0 meters