A pole 5 m high is fixed on the top of a tower. The angle of elevation of the top of the pole as observed from a point A on the ground is `60^(@)` and the angle of depression of the point A from the top of the tower is `45^(@)`. Find the height of the tower.

Answer

To solve the problem step by step, we will use trigonometric concepts related to angles of elevation and depression.

### Step 1: Understand the Problem
We have a tower with a height we need to find, and on top of this tower, there is a pole that is 5 meters high. We denote the height of the tower as \( h \). The total height from the ground to the top of the pole is \( h + 5 \).

### Step 2: Set Up the Diagram
1. Let point \( A \) be the point on the ground from where the angles are measured.
2. Let point \( B \) be the top of the tower.
3. Let point \( C \) be the top of the pole.
4. The height of the tower is \( h \), and the height of the pole is 5 m.

### Step 3: Use the Angle of Elevation
From point \( A \), the angle of elevation to point \( C \) (the top of the pole) is \( 60^\circ \). We can use the tangent function:
\[ \tan(60^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{h + 5}{d} \]
where \( d \) is the horizontal distance from point \( A \) to the base of the tower.

Since \( \tan(60^\circ) = \sqrt{3} \), we have:
\[ \sqrt{3} = \frac{h + 5}{d} \quad \text{(Equation 1)} \]

### Step 4: Use the Angle of Depression
From point \( B \) (the top of the tower), the angle of depression to point \( A \) is \( 45^\circ \). Using the tangent function again:
\[ \tan(45^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{h}{d} \]
Since \( \tan(45^\circ) = 1 \), we have:
\[ 1 = \frac{h}{d} \quad \text{(Equation 2)} \]
This implies:
\[ d = h \quad \text{(Equation 3)} \]

### Step 5: Substitute Equation 3 into Equation 1
Now, we substitute \( d \) from Equation 3 into Equation 1:
\[ \sqrt{3} = \frac{h + 5}{h} \]

### Step 6: Solve for \( h \)
Cross-multiplying gives:
\[ \sqrt{3}h = h + 5 \]
Rearranging the equation:
\[ \sqrt{3}h - h = 5 \]
Factoring out \( h \):
\[ h(\sqrt{3} - 1) = 5 \]
Thus, we can solve for \( h \):
\[ h = \frac{5}{\sqrt{3} - 1} \]

### Step 7: Rationalize the Denominator
To rationalize the denominator:
\[ h = \frac{5(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{5(\sqrt{3} + 1)}{3 - 1} = \frac{5(\sqrt{3} + 1)}{2} \]

### Final Answer
The height of the tower is:
\[ h = \frac{5(\sqrt{3} + 1)}{2} \text{ meters} \]