The angle of elevation of the top of a tower is observed to be `60^(@)` . At a point , 30 m vertically above the first point of observation, the elevation is found to be `45^(@)` . Find :
its horizontal distance from the points of observation.

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the problem and draw a diagram
We have a tower and two points of observation. The first point (let's call it A) is at the base of the tower, and the second point (let's call it B) is 30 meters above point A. The angle of elevation from point A to the top of the tower (point D) is 60 degrees, and the angle of elevation from point B to point D is 45 degrees.

### Step 2: Define the variables
Let:
- \( h \) = height of the tower (AD)
- \( x \) = horizontal distance from the foot of the tower (point C) to point A (BD)

### Step 3: Set up the equations using trigonometry
From triangle ABD (where angle ADB = 60 degrees):
\[ \tan(60^\circ) = \frac{h}{x} \]
Since \( \tan(60^\circ) = \sqrt{3} \), we can write:
\[ \sqrt{3} = \frac{h}{x} \implies h = x\sqrt{3} \quad \text{(Equation 1)} \]

From triangle ABE (where angle ABE = 45 degrees):
\[ \tan(45^\circ) = \frac{AE}{EC} \]
Here, \( AE = h - 30 \) (since point B is 30 meters above point A), and \( EC = x \). Thus:
\[ 1 = \frac{h - 30}{x} \implies h - 30 = x \quad \text{(Equation 2)} \]

### Step 4: Solve the equations
Now we have two equations:
1. \( h = x\sqrt{3} \)
2. \( h = x + 30 \)

We can set these equal to each other:
\[ x\sqrt{3} = x + 30 \]

Rearranging gives:
\[ x\sqrt{3} - x = 30 \]
\[ x(\sqrt{3} - 1) = 30 \]
\[ x = \frac{30}{\sqrt{3} - 1} \]

### Step 5: Rationalize the denominator
To simplify \( x \):
\[ x = \frac{30(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{30(\sqrt{3} + 1)}{3 - 1} = \frac{30(\sqrt{3} + 1)}{2} \]
\[ x = 15(\sqrt{3} + 1) \]

### Step 6: Calculate the numerical value
Using \( \sqrt{3} \approx 1.732 \):
\[ x \approx 15(1.732 + 1) = 15(2.732) \approx 40.98 \text{ meters} \]

### Final Answer
The horizontal distance from the point of observation to the foot of the tower is approximately **40.98 meters**.

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