From the top of a 20 metre high building , the angle of elevation of the top of a tower is `60^(@)` and the angle of depression of its foot is at `45^(@)` , then the height of the tower is `(sqrt(3)=1.732)`

Options

A. a.45 . 46 metre
B. b.45 . 64 metre
C. c.54 . 64 metre
D. d.54 . 46 metre

Answer

To solve the problem, we need to find the height of the tower based on the given angles of elevation and depression from the top of a 20-meter high building. Let's break it down step by step.

### Step 1: Understand the Setup
We have a building of height \( BC = 20 \) meters. From the top of this building, we have:
- An angle of elevation of \( 60^\circ \) to the top of the tower (let's call it point T).
- An angle of depression of \( 45^\circ \) to the foot of the tower (let's call it point D).

### Step 2: Use the Angle of Depression
From the top of the building (point B), we can analyze triangle \( BCD \) where:
- \( \angle BDC = 45^\circ \)
- \( BC = 20 \) meters (height of the building)
- \( BD \) is the horizontal distance from the building to the foot of the tower.

Using the tangent function for \( \angle BDC \):
\[ \tan(45^\circ) = \frac{BC}{BD} \]
Since \( \tan(45^\circ) = 1 \):
\[ 1 = \frac{20}{BD} \]
Thus, we find:
\[ BD = 20 \text{ meters} \]

### Step 3: Use the Angle of Elevation
Now we analyze triangle \( BCT \) where:
- \( \angle BTC = 60^\circ \)
- \( BC = 20 \) meters (height of the building)
- \( BT \) is the height of the tower above the building.

Using the tangent function for \( \angle BTC \):
\[ \tan(60^\circ) = \frac{BT}{BD} \]
Since \( \tan(60^\circ) = \sqrt{3} \):
\[ \sqrt{3} = \frac{BT}{20} \]
Thus, we find:
\[ BT = 20\sqrt{3} \text{ meters} \]

### Step 4: Calculate the Total Height of the Tower
The total height of the tower \( HT \) is the sum of the height of the building and the height of the tower above the building:
\[ HT = BC + BT = 20 + 20\sqrt{3} \]
Substituting \( \sqrt{3} \approx 1.732 \):
\[ HT = 20 + 20 \times 1.732 = 20 + 34.64 = 54.64 \text{ meters} \]

### Final Answer
The height of the tower is \( 54.64 \) meters.

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