The angle of elevation of the top of an unfinished tower from a point at a distance of 80 m from its base is `30^(@)` . How much higher must the tower be raised so that its angle of elevation as the same point may be `60^(@)`

Answer

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

### Step 1: Understand the Problem
We have a tower (AB) and a point (C) at a distance of 80 m from the base (A) of the tower. The angle of elevation from point C to the top of the tower is 30 degrees. We need to find out how much higher the tower must be raised so that the angle of elevation from point C becomes 60 degrees.

### Step 2: Set Up the First Triangle
In triangle ABC, we have:
- Angle ACB = 30 degrees
- AC = 80 m (distance from the point to the base of the tower)
- AB = height of the tower (let's denote it as h1)

Using the tangent function:
\[ \tan(30^\circ) = \frac{AB}{AC} = \frac{h_1}{80} \]
We know that \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\).

### Step 3: Solve for h1
Using the equation from Step 2:
\[ \frac{1}{\sqrt{3}} = \frac{h_1}{80} \]
Cross-multiplying gives:
\[ h_1 = \frac{80}{\sqrt{3}} \approx 46.19 \text{ m} \]

### Step 4: Set Up the Second Triangle
Now, we consider the new height of the tower (let's denote it as h2) when the angle of elevation is 60 degrees. In triangle DEF:
- Angle DFE = 60 degrees
- EF = 80 m (the same distance)

Using the tangent function again:
\[ \tan(60^\circ) = \frac{DE}{EF} = \frac{h_2}{80} \]
We know that \(\tan(60^\circ) = \sqrt{3}\).

### Step 5: Solve for h2
Using the equation from Step 4:
\[ \sqrt{3} = \frac{h_2}{80} \]
Cross-multiplying gives:
\[ h_2 = 80\sqrt{3} \approx 138.56 \text{ m} \]

### Step 6: Calculate the Height Increase
To find out how much higher the tower must be raised, we subtract the original height from the new height:
\[ \text{Height increase} = h_2 - h_1 = 80\sqrt{3} - \frac{80}{\sqrt{3}} \]
Calculating this gives:
\[ \text{Height increase} = 138.56 - 46.19 \approx 92.37 \text{ m} \]

### Final Answer
The tower must be raised by approximately **92.37 meters**.

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