The angle of elevation of the top of an unfinished pillar at a point 150 metres from its base is `30^(@)` The height (in metres) that the pillar must be raised so that its angle of elevation at the same point may be `45^(@)` , is (Take, `sqrt(3)=1.732)`

Options

A. `63.4`
B. `86.6`
C. `126.8`
D. `173.2`

Answer

To solve the problem step by step, we will use trigonometric principles, specifically the tangent function, which relates the angle of elevation to the height and distance from the base of the pillar.

### Step 1: Understand the Problem
We have a pillar and a point 150 meters away from its base. The angle of elevation to the top of the pillar from this point is initially 30 degrees. We need to find out how much higher the pillar must be raised so that the angle of elevation becomes 45 degrees.

### Step 2: Calculate the Initial Height of the Pillar
Using the tangent function for the angle of elevation of 30 degrees:
\[ \tan(30^\circ) = \frac{\text{Height of the pillar (h)}}{\text{Distance from the base (150 m)}} \]
We know that:
\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \]
Thus, we can set up the equation:
\[ \frac{1}{\sqrt{3}} = \frac{h}{150} \]
Rearranging gives:
\[ h = 150 \times \frac{1}{\sqrt{3}} = \frac{150}{\sqrt{3}} \]
To simplify, we multiply the numerator and denominator by \(\sqrt{3}\):
\[ h = \frac{150\sqrt{3}}{3} = 50\sqrt{3} \text{ meters} \]

### Step 3: Set Up the Equation for the New Height
Now, we want the angle of elevation to be 45 degrees. Using the tangent function again:
\[ \tan(45^\circ) = \frac{\text{Height of the pillar (h + x)}}{\text{Distance from the base (150 m)}} \]
Since \(\tan(45^\circ) = 1\), we have:
\[ 1 = \frac{h + x}{150} \]
Rearranging gives:
\[ h + x = 150 \]
Thus:
\[ x = 150 - h \]

### Step 4: Substitute the Value of h
Substituting the value of \(h\) we found earlier:
\[ x = 150 - 50\sqrt{3} \]

### Step 5: Calculate the Value of x
Now, substituting \(\sqrt{3} \approx 1.732\):
\[ x = 150 - 50 \times 1.732 \]
Calculating \(50 \times 1.732\):
\[ 50 \times 1.732 = 86.6 \]
Thus:
\[ x = 150 - 86.6 = 63.4 \text{ meters} \]

### Final Answer
The height that the pillar must be raised so that its angle of elevation at the same point may be \(45^\circ\) is **63.4 meters**.

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