The angle of elevation of the top of a tower at a point on the ground is `30^(@)`. If the height of the tower is tripled, find the angle of elevation of the top of the same point.

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 a tower of height \( H \) and the angle of elevation from a point on the ground to the top of the tower is \( 30^\circ \). We need to find the new angle of elevation when the height of the tower is tripled (i.e., the new height is \( 3H \)).

### Step 2: Set Up the First Triangle
Let’s denote:
- \( A \) as the top of the tower,
- \( B \) as the base of the tower,
- \( C \) as the point on the ground from where the angle of elevation is measured.

From triangle \( ABC \):
- The height of the tower \( AB = H \).
- The distance from point \( C \) to the base of the tower \( BC = D \).
- The angle \( \angle ACB = 30^\circ \).

Using the tangent function:
\[ \tan(30^\circ) = \frac{AB}{BC} = \frac{H}{D} \]

### Step 3: Calculate \( D \) in Terms of \( H \)
We know that:
\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \]
Thus, we can write:
\[ \frac{1}{\sqrt{3}} = \frac{H}{D} \]
Cross-multiplying gives:
\[ H = \frac{D}{\sqrt{3}} \]

### Step 4: Set Up the Second Triangle
Now, when the height of the tower is tripled, the new height \( AB' = 3H \). We need to find the new angle of elevation \( \theta \) from point \( C \) to the top of the new tower \( A' \).

In triangle \( A'BC \):
- The height \( A'B = 3H \).
- The distance \( BC = D \) remains the same.

Using the tangent function again:
\[ \tan(\theta) = \frac{A'B}{BC} = \frac{3H}{D} \]

### Step 5: Substitute \( H \) into the Equation
Substituting \( H \) from our earlier calculation:
\[ \tan(\theta) = \frac{3 \left(\frac{D}{\sqrt{3}}\right)}{D} \]
This simplifies to:
\[ \tan(\theta) = \frac{3}{\sqrt{3}} = \sqrt{3} \]

### Step 6: Find \( \theta \)
Now, we need to find the angle \( \theta \) such that:
\[ \tan(\theta) = \sqrt{3} \]
From trigonometric values, we know:
\[ \theta = 60^\circ \]

### Final Answer
Thus, the angle of elevation when the height of the tower is tripled is:
\[ \theta = 60^\circ \]

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