An observer 1.5 m tall is `20sqrt(3)` m away from a chimney. The angle of elevation from the top of the chimney from his eyes is 30° and from bottom is 45°. Find the height of the chimney.

Answer

To find the height of the chimney, we will use the information provided about the observer's height, the distance from the chimney, and the angles of elevation.

### Step 1: Understand the setup
- The observer is 1.5 m tall.
- The distance from the observer to the chimney is \(20\sqrt{3}\) m.
- The angle of elevation to the top of the chimney from the observer's eyes is \(30^\circ\).
- The angle of elevation to the bottom of the chimney from the observer's eyes is \(45^\circ\).

### Step 2: Define the variables
Let:
- \(H\) = height of the chimney
- The height of the observer's eyes from the ground = 1.5 m
- The height of the chimney above the observer's eyes = \(H - 1.5\) m

### Step 3: Use the angle of elevation to find the height of the chimney above the observer's eyes
From the observer's eyes to the top of the chimney, we can use the tangent of the angle of elevation:
\[ \tan(30^\circ) = \frac{H - 1.5}{20\sqrt{3}} \]
We know that \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\), so we can substitute this value:
\[ \frac{1}{\sqrt{3}} = \frac{H - 1.5}{20\sqrt{3}} \]

### Step 4: Cross-multiply to solve for \(H - 1.5\)
Cross-multiplying gives:
\[ 1 \cdot 20\sqrt{3} = (H - 1.5) \cdot \sqrt{3} \]
This simplifies to:
\[ 20\sqrt{3} = (H - 1.5)\sqrt{3} \]

### Step 5: Divide both sides by \(\sqrt{3}\)
\[ 20 = H - 1.5 \]

### Step 6: Solve for \(H\)
Adding 1.5 to both sides:
\[ H = 20 + 1.5 = 21.5 \text{ m} \]

### Conclusion
The total height of the chimney is \(21.5\) m.