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 :
the height of the tower

Answer

To solve the problem step by step, we will use trigonometric ratios and the information given in the question.

### Step 1: Understand the Problem
We have a tower (let's denote its height as \( h \)). The angle of elevation from point D (the first observation point) to the top of the tower is \( 60^\circ \). From point E (which is 30 m above point D), the angle of elevation to the top of the tower is \( 45^\circ \).

### Step 2: Set Up the Diagram
Let:
- \( A \) be the top of the tower.
- \( B \) be the base of the tower.
- \( D \) be the first observation point.
- \( E \) be the second observation point (30 m above D).
- \( h \) be the height of the tower \( AB \).
- \( DE = 30 \) m (the vertical distance between D and E).

### Step 3: Use Trigonometric Ratios
From point E, the angle of elevation to the top of the tower is \( 45^\circ \):
\[ \tan(45^\circ) = \frac{h}{EB} \]
Since \( \tan(45^\circ) = 1 \):
\[ 1 = \frac{h}{EB} \implies h = EB \quad \text{(Equation 1)} \]

From point D, the angle of elevation to the top of the tower is \( 60^\circ \):
\[ \tan(60^\circ) = \frac{h + 30}{DC} \]
Since \( \tan(60^\circ) = \sqrt{3} \):
\[ \sqrt{3} = \frac{h + 30}{EB} \]
Rearranging gives:
\[ EB \cdot \sqrt{3} = h + 30 \quad \text{(Equation 2)} \]

### Step 4: Substitute Equation 1 into Equation 2
From Equation 1, we know \( EB = h \). Substituting this into Equation 2:
\[ h \cdot \sqrt{3} = h + 30 \]
Rearranging gives:
\[ h \cdot \sqrt{3} - h = 30 \]
Factoring out \( h \):
\[ h(\sqrt{3} - 1) = 30 \]
Thus:
\[ h = \frac{30}{\sqrt{3} - 1} \]

### Step 5: Rationalize the Denominator
To simplify \( h \):
\[ h = \frac{30(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{30(\sqrt{3} + 1)}{3 - 1} = \frac{30(\sqrt{3} + 1)}{2} \]
Calculating this gives:
\[ h = 15(\sqrt{3} + 1) \approx 15(1.732 + 1) \approx 15(2.732) \approx 40.98 \text{ m} \]

### Step 6: Find the Total Height of the Tower
The total height of the tower \( AC \) is:
\[ AC = h + 30 = 40.98 + 30 = 70.98 \text{ m} \]

### Final Answer
The height of the tower is approximately \( 70.98 \) meters.

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