Diagram illustrating the calculation of tower height using angles of elevation from two different distances.

You're absolutely right to start by drawing a diagram — it’s the best way to visualize and solve this problem! Since no image was provided, let’s reconstruct the scenario based on your description.

---

📌 Problem Summary:

- A person looks at the top of a tower from two different positions.
- First position (Point C): Angle of elevation = 21.4°
- Second position (Point D): Person moves 85.3 meters farther away → angle of elevation = 16.7°
- We need to find the height of the tower, which we’ll call x.
- Let the horizontal distance from the base of the tower (B) to the first position (C) be y.
- Then, the distance from B to D is y + 85.3.

---

🧭 Step-by-step Solution:

We’ll use right-triangle trigonometry (tangent function), since we have angles and opposite/adjacent sides.

#### Triangle ABC:
- Right triangle with right angle at B.
- Angle at C = 21.4°
- Opposite side to angle = height of tower = x
- Adjacent side = y

So,

> tan(21.4°) = x / y
→ Equation (1): x = y · tan(21.4°)

#### Triangle ABD:
- Right triangle with right angle at B.
- Angle at D = 16.7°
- Opposite side = x
- Adjacent side = y + 85.3

So,

> tan(16.7°) = x / (y + 85.3)
→ Equation (2): x = (y + 85.3) · tan(16.7°)

---

🔗 Set Equations Equal:

Since both equal x, set them equal:

> y · tan(21.4°) = (y + 85.3) · tan(16.7°)

Let’s compute the tangent values (use calculator in degree mode):

- tan(21.4°) ≈ 0.3910
- tan(16.7°) ≈ 0.3000

*(Note: These are approximate — we’ll keep more decimals for accuracy.)*

Actually, let’s use more precise values:

- tan(21.4°) = tan(21.4) ≈ 0.391043
- tan(16.7°) = tan(16.7) ≈ 0.299877

Now plug in:

> y × 0.391043 = (y + 85.3) × 0.299877

Expand the right-hand side:

> 0.391043y = 0.299877y + 0.299877 × 85.3

Compute the constant term:

> 0.299877 × 85.3 ≈ 25.580

So:

> 0.391043y - 0.299877y = 25.580

> 0.091166y = 25.580

Now solve for y:

> y = 25.580 / 0.091166 ≈ 280.57 meters

---

📏 Now Find Height x:

Use Equation (1): x = y · tan(21.4°)

> x = 280.57 × 0.391043 ≈ 109.72 meters

---

✔ Final Answer:

> The height of the tower is approximately 109.7 meters.

---

🧠 Verification (Optional but Good Practice):

Check with second equation:

> x = (y + 85.3) × tan(16.7°)
= (280.57 + 85.3) × 0.299877
= 365.87 × 0.299877 ≈ 109.72 meters

✔ Matches! So our answer is consistent.

---

🎯 Conclusion:

By setting up two right triangles and using the tangent function, we related the unknown height to the unknown base distances, then solved the system algebraically. The key insight was recognizing that the height remains constant while the horizontal distance changes — allowing us to equate the two expressions for height.

---

✔ Answer: The height of the tower is approximately 109.7 meters.