The angle of elevation of the top of a tower from a point P on the ground is `alpha`. After walking a distance d meter towards the foot of the tower, angle of elevation is found to be `beta`. Which angle of elevation is greater?

Answer

To determine which angle of elevation is greater, we will analyze the situation using trigonometric relationships. Let's break down the problem step by step.

### Step 1: Understanding the Setup
Let:
- \( AB \) be the height of the tower.
- \( P \) be the initial point on the ground from where the angle of elevation is \( \alpha \).
- \( C \) be the foot of the tower.
- \( D \) be the point after walking \( d \) meters towards the tower, where the angle of elevation is \( \beta \).

### Step 2: Establishing Relationships Using Trigonometry
From point \( P \):
- The angle of elevation \( \alpha \) gives us the relationship:
\[ \tan(\alpha) = \frac{AB}{BC} \]
where \( BC \) is the horizontal distance from point \( P \) to the foot of the tower \( C \).

From point \( D \):
- The angle of elevation \( \beta \) gives us the relationship:
\[ \tan(\beta) = \frac{AB}{BD} \]
where \( BD \) is the horizontal distance from point \( D \) to the foot of the tower \( C \).

### Step 3: Relating the Distances
Since point \( D \) is \( d \) meters closer to the tower than point \( P \), we can express the distances as:
\[ BD = BC - d \]

### Step 4: Comparing the Tangents
Now, we can rewrite the equations for \( \tan(\alpha) \) and \( \tan(\beta) \):
1. From point \( P \):
\[ \tan(\alpha) = \frac{AB}{BC} \]
2. From point \( D \):
\[ \tan(\beta) = \frac{AB}{BC - d} \]

### Step 5: Analyzing the Inequality
To find out which angle is greater, we need to compare \( \tan(\alpha) \) and \( \tan(\beta) \):
- Since \( BC > BD \) (because \( D \) is closer to the tower), we have:
\[ BC > BC - d \]
- This implies:
\[ \frac{AB}{BC} < \frac{AB}{BC - d} \]
- Therefore:
\[ \tan(\alpha) < \tan(\beta) \]

### Step 6: Conclusion
Since \( \tan(\alpha) < \tan(\beta) \), it follows that:
\[ \alpha < \beta \]
Thus, the angle of elevation \( \beta \) is greater than \( \alpha \).