The angle of elevation of a tower at a point is `45^(@)`. After going 40 m towards the foot of the tower, the angle of elevation of the tower becomes `60^(@)`. Find the height of the tower.

Answer

To find the height of the tower based on the given angles of elevation, we can follow these steps:

### Step 1: Understand the Problem
We have two points of observation:
- Point P, where the angle of elevation to the top of the tower (A) is 45 degrees.
- Point Q, which is 40 meters closer to the tower, where the angle of elevation becomes 60 degrees.

### Step 2: Set Up the Diagram
Let's denote:
- The height of the tower as \( h \).
- The distance from point P to the foot of the tower (B) as \( x \).
- The distance from point Q to the foot of the tower (B) is \( x - 40 \).

### Step 3: Use Trigonometry for Point Q
From point Q, we can use the tangent function:
\[ \tan(60^\circ) = \frac{AB}{BQ} \]
Substituting the known values:
\[ \tan(60^\circ) = \sqrt{3} \quad \text{and} \quad BQ = x - 40 \]
Thus, we have:
\[ \sqrt{3} = \frac{h}{x - 40} \]
From this, we can express \( h \):
\[ h = \sqrt{3}(x - 40) \quad \text{(Equation 1)} \]

### Step 4: Use Trigonometry for Point P
From point P, we again use the tangent function:
\[ \tan(45^\circ) = \frac{AB}{BP} \]
Since \( \tan(45^\circ) = 1 \):
\[ 1 = \frac{h}{x} \]
Thus, we can express \( h \) as:
\[ h = x \quad \text{(Equation 2)} \]

### Step 5: Equate the Two Expressions for \( h \)
From Equation 1 and Equation 2, we have:
\[ \sqrt{3}(x - 40) = x \]

### Step 6: Solve for \( x \)
Rearranging the equation:
\[ \sqrt{3}x - 40\sqrt{3} = x \]
\[ \sqrt{3}x - x = 40\sqrt{3} \]
Factoring out \( x \):
\[ x(\sqrt{3} - 1) = 40\sqrt{3} \]
Thus, we can solve for \( x \):
\[ x = \frac{40\sqrt{3}}{\sqrt{3} - 1} \]

### Step 7: Substitute \( x \) Back to Find \( h \)
Now substituting \( x \) back into Equation 2:
\[ h = x = \frac{40\sqrt{3}}{\sqrt{3} - 1} \]

### Step 8: Rationalize the Denominator
To simplify \( h \):
\[ h = \frac{40\sqrt{3}(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{40\sqrt{3}(\sqrt{3} + 1)}{2} = 20\sqrt{3}(\sqrt{3} + 1) \]
Thus, the height of the tower \( h \) is:
\[ h = 20(3 + \sqrt{3}) = 60 + 20\sqrt{3} \]

### Final Answer
The height of the tower is \( 20(3 + \sqrt{3}) \) meters.

---