The angle of elevation of the top of an unfinished pillar at a point 150 m from its base is `30^@`. If the angle of elevation at the same point is to be `45^@`, then the pillar has to be raised to a height of how many metres?

Options

A. 59.4 m
B. 61.4 m
C. 62.4 m
D. 63.4 m

Answer

To solve the problem step by step, we will use trigonometric principles to find the height of the pillar that needs to be raised.

### Given:
- Distance from the base of the pillar (BD) = 150 m
- Angle of elevation (∠ABD) = 30°
- New angle of elevation (∠DBC) = 45°

### Step 1: Find the height of the pillar (AB) when the angle of elevation is 30°.
Using the tangent function:
\[ \tan(30°) = \frac{AB}{BD} \]
Where:
- \(AB\) is the height of the pillar,
- \(BD\) is the distance from the point to the base of the pillar (150 m).

From trigonometric values, we know:
\[ \tan(30°) = \frac{1}{\sqrt{3}} \]
Substituting the values:
\[ \frac{1}{\sqrt{3}} = \frac{AB}{150} \]

### Step 2: Solve for AB.
Rearranging the equation gives:
\[ AB = 150 \cdot \frac{1}{\sqrt{3}} = \frac{150}{\sqrt{3}} \approx 86.6 \text{ m} \]

### Step 3: Find the height of the pillar (BC) when the angle of elevation is 45°.
Using the tangent function again:
\[ \tan(45°) = \frac{BC}{BD} \]
Where:
- \(BC\) is the new height of the pillar.

Since \(\tan(45°) = 1\), we have:
\[ 1 = \frac{BC}{150} \]

### Step 4: Solve for BC.
Rearranging gives:
\[ BC = 150 \text{ m} \]

### Step 5: Find the height to which the pillar needs to be raised (AC).
The height to which the pillar needs to be raised is given by:
\[ AC = BC - AB \]
Substituting the values we found:
\[ AC = 150 - \frac{150}{\sqrt{3}} = 150 \left(1 - \frac{1}{\sqrt{3}}\right) \]

### Step 6: Simplify AC.
To simplify:
\[ AC = 150 \left(\frac{\sqrt{3} - 1}{\sqrt{3}}\right) \]
Now, rationalizing:
\[ AC = 150 \cdot \frac{\sqrt{3} - 1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = 150 \cdot \frac{3 - \sqrt{3}}{3} \]

### Step 7: Calculate AC.
\[ AC = 50(3 - \sqrt{3}) \approx 50(3 - 1.732) \approx 50(1.268) \approx 63.4 \text{ m} \]

### Final Answer:
The pillar has to be raised by approximately **63.4 meters**.

---