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. 61.4m
C. 62.4m
D. 63.4m

Answer

To solve the problem step by step, we will use trigonometric principles related to angles of elevation.

### Step 1: Understand the Problem
We have an unfinished pillar and we want to find out how much higher it needs to be raised so that the angle of elevation from a point 150 meters away from its base changes from 30 degrees to 45 degrees.

### Step 2: Set Up the Diagram
Let:
- \( A \) be the base of the pillar.
- \( B \) be the top of the unfinished pillar.
- \( C \) be the point from which the angle of elevation is measured (150 m from the base).
- The height of the unfinished pillar is \( h \).

### Step 3: Calculate the Height of the Pillar at 30 Degrees
Using the angle of elevation of 30 degrees, we can use the tangent function:
\[ \tan(30^\circ) = \frac{h}{150} \]
We know that \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \). Therefore, we can write:
\[ \frac{1}{\sqrt{3}} = \frac{h}{150} \]

### Step 4: Solve for \( h \)
Cross-multiplying gives:
\[ h = 150 \cdot \frac{1}{\sqrt{3}} = \frac{150}{\sqrt{3}} \approx 86.6 \text{ m} \]

### Step 5: Calculate the Height of the Pillar at 45 Degrees
Next, we consider the angle of elevation of 45 degrees:
\[ \tan(45^\circ) = \frac{H}{150} \]
Since \( \tan(45^\circ) = 1 \), we have:
\[ 1 = \frac{H}{150} \]
This implies:
\[ H = 150 \text{ m} \]

### Step 6: Find the Increase in Height
Now, we need to find out how much the height of the pillar needs to be increased:
\[ \text{Increase in height} = H - h = 150 - \frac{150}{\sqrt{3}} \]

### Step 7: Simplify the Expression
To simplify:
\[ \text{Increase in height} = 150 \left(1 - \frac{1}{\sqrt{3}}\right) \]
\[ = 150 \left(\frac{\sqrt{3} - 1}{\sqrt{3}}\right) \]

### Step 8: Rationalize the Denominator
Multiply numerator and denominator by \( \sqrt{3} \):
\[ = 150 \cdot \frac{\sqrt{3}(\sqrt{3} - 1)}{3} = 50\sqrt{3}(\sqrt{3} - 1) \]

### Step 9: Calculate the Final Value
Now, substituting \( \sqrt{3} \approx 1.732 \):
\[ \text{Increase in height} \approx 50 \cdot 1.732 \cdot (1.732 - 1) \approx 50 \cdot 1.732 \cdot 0.732 \approx 63.4 \text{ m} \]

### Conclusion
Thus, the pillar needs to be raised by approximately **63.4 meters**.

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