Find the height of a building, when it is found that on walking towards it 40 m in a horizontal line through its base the angular elevation of its top changes from `30^(@) t o 45^(@)`

Answer

To find the height of the building, we can use trigonometric ratios based on the angles of elevation given in the problem. Let's break down the solution step by step.

### Step 1: Understand the Problem
We have a building of height \( h \). When standing at point \( D \), the angle of elevation to the top of the building \( B \) is \( 30^\circ \). After walking \( 40 \, m \) towards the building to point \( C \), the angle of elevation changes to \( 45^\circ \).

### Step 2: Set Up the Diagram
Let:
- \( A \) be the base of the building,
- \( B \) be the top of the building,
- \( D \) be the initial position,
- \( C \) be the position after walking \( 40 \, m \).

We have:
- \( AD = 40 + BC \) (the distance from point \( D \) to point \( A \)),
- \( BC = h \) (the height of the building).

### Step 3: Use Trigonometric Ratios
1. **From point C** (where the angle of elevation is \( 45^\circ \)):
\[ \tan(45^\circ) = \frac{h}{BC} \]
Since \( \tan(45^\circ) = 1 \):
\[ 1 = \frac{h}{BC} \implies BC = h \quad \text{(Equation 1)} \]

2. **From point D** (where the angle of elevation is \( 30^\circ \)):
\[ \tan(30^\circ) = \frac{h}{AD} = \frac{h}{40 + h} \]
Since \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \):
\[ \frac{1}{\sqrt{3}} = \frac{h}{40 + h} \]

### Step 4: Solve for h
Cross-multiplying gives:
\[ h \cdot \sqrt{3} = 40 + h \]
Rearranging this equation:
\[ h \sqrt{3} - h = 40 \]
Factoring out \( h \):
\[ h(\sqrt{3} - 1) = 40 \]
Thus,
\[ h = \frac{40}{\sqrt{3} - 1} \]

### Step 5: Rationalize the Denominator
To simplify \( h \):
\[ h = \frac{40(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{40(\sqrt{3} + 1)}{3 - 1} = \frac{40(\sqrt{3} + 1)}{2} = 20(\sqrt{3} + 1) \]

### Step 6: Calculate the Height
Now, substituting the value of \( \sqrt{3} \approx 1.732 \):
\[ h \approx 20(1.732 + 1) = 20(2.732) \approx 54.64 \, m \]

### Final Answer
The height of the building is approximately \( 54.64 \, m \).

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