The angle of elevation of the top of a half constructed tower at a distance of 150 m from the base of the tower is `30^(@)`. If angle of elevation of top of tower from that point is to be `45^(@)`, then by what amount height of the tower be increased ?

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 ratios to find the height of the half-constructed tower and then the complete height of the tower. Finally, we will determine the increase in height.

### Step 1: Understanding the Problem
We have a half-constructed tower, and we need to find the height increase required to complete the tower. We know the angles of elevation and the distance from the tower.

### Step 2: Draw the Diagram
1. Let point A be the top of the half-constructed tower.
2. Let point B be the base of the tower.
3. Let point C be the point from where the angles of elevation are measured, which is 150 m away from point B.
4. The angle of elevation to point A (the top of the half-constructed tower) is 30°.
5. The angle of elevation to point D (the top of the complete tower) is 45°.

### Step 3: Calculate the Height of the Half-Constructed Tower (AB)
Using the triangle ABC where:
- Angle CAB = 30°
- Distance BC = 150 m

Using the tangent function:
\[ \tan(30°) = \frac{AB}{BC} \]
\[ \tan(30°) = \frac{AB}{150} \]
Since \(\tan(30°) = \frac{1}{\sqrt{3}}\):
\[ \frac{1}{\sqrt{3}} = \frac{AB}{150} \]
Cross-multiplying gives:
\[ AB = \frac{150}{\sqrt{3}} \quad \text{(1)} \]

### Step 4: Calculate the Height of the Complete Tower (AD)
Using the triangle BCD where:
- Angle DBC = 45°
- Distance BC = 150 m

Using the tangent function:
\[ \tan(45°) = \frac{AD}{BC} \]
\[ \tan(45°) = \frac{AD}{150} \]
Since \(\tan(45°) = 1\):
\[ 1 = \frac{AD}{150} \]
Cross-multiplying gives:
\[ AD = 150 \quad \text{(2)} \]

### Step 5: Find the Increase in Height
Now, we need to find the increase in height:
\[ \text{Increase in height} = AD - AB \]
Substituting the values from (1) and (2):
\[ \text{Increase in height} = 150 - \frac{150}{\sqrt{3}} \]
To simplify, we can find a common denominator:
\[ = 150 \left(1 - \frac{1}{\sqrt{3}}\right) \]
\[ = 150 \left(\frac{\sqrt{3} - 1}{\sqrt{3}}\right) \]
\[ = \frac{150(\sqrt{3} - 1)}{\sqrt{3}} \quad \text{(3)} \]

### Step 6: Calculate the Numerical Value
Now, we can calculate the numerical value:
Using \(\sqrt{3} \approx 1.732\):
\[ = \frac{150(1.732 - 1)}{1.732} \]
\[ = \frac{150(0.732)}{1.732} \]
Calculating this gives:
\[ \approx \frac{109.8}{1.732} \approx 63.4 \text{ m} \]

### Final Answer
The increase in height of the tower is approximately **63.4 meters**.

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