The shadow of a tower standing on a level ground is found to be 60 metres longer when the sun's altitude is `30^(@)` than when it is `45^(@)`. The height of the tower is____

Answer

To solve the problem step by step, we can follow these calculations:

### Step 1: Understand the problem
We have a tower whose height we need to find. The shadow of the tower is longer when the sun's altitude is at 30 degrees compared to when it is at 45 degrees. The difference in the lengths of the shadows is given as 60 meters.

### Step 2: Set up the equations
Let the height of the tower be \( h \) meters and the length of the shadow when the sun's altitude is 45 degrees be \( x \) meters.

From trigonometry, we know:
- When the angle of elevation is 45 degrees:
\[ \tan(45^\circ) = \frac{h}{x} \implies h = x \]

- When the angle of elevation is 30 degrees:
\[ \tan(30^\circ) = \frac{h}{60 + x} \implies h = (60 + x) \cdot \tan(30^\circ) = (60 + x) \cdot \frac{1}{\sqrt{3}} \]

### Step 3: Substitute \( h \) from the first equation into the second equation
From the first equation, we have \( h = x \). Substitute this into the second equation:
\[ x = (60 + x) \cdot \frac{1}{\sqrt{3}} \]

### Step 4: Solve for \( x \)
Multiply both sides by \( \sqrt{3} \):
\[ x \sqrt{3} = 60 + x \]
Rearranging gives:
\[ x \sqrt{3} - x = 60 \]
Factoring out \( x \):
\[ x(\sqrt{3} - 1) = 60 \]
Thus,
\[ x = \frac{60}{\sqrt{3} - 1} \]

### Step 5: Rationalize the denominator
To rationalize the denominator, multiply the numerator and denominator by \( \sqrt{3} + 1 \):
\[ x = \frac{60(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{60(\sqrt{3} + 1)}{3 - 1} = \frac{60(\sqrt{3} + 1)}{2} = 30(\sqrt{3} + 1) \]

### Step 6: Find the height \( h \)
Since \( h = x \), we have:
\[ h = 30(\sqrt{3} + 1) \text{ meters} \]

### Final Answer
The height of the tower is \( 30(\sqrt{3} + 1) \) meters.

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