From the top of a 12 m high building, the angle of elevation of the top of a tower is `60^@` and the angle of depression of the foot of the tower is `theta` , such that `tan theta = (3)/(4)` What is the height of the tower `(sqrt(3) = 1.73)` ?

Options

A. 37.95 m
B. 39.68 m
C. 41.41 m
D. 36.22 m

Answer

To solve the problem step by step, we will follow the given information and apply trigonometric principles.

### Step 1: Understand the Problem
We have a building of height 12 m. From the top of this building, we know the angle of elevation to the top of a tower is \(60^\circ\) and the angle of depression to the foot of the tower is \(\theta\), where \(\tan \theta = \frac{3}{4}\). We need to find the height of the tower.

### Step 2: Draw a Diagram
Draw a diagram to visualize the situation:
- Let point A be the top of the building (12 m high).
- Let point B be the foot of the building.
- Let point P be the top of the tower.
- Let point Q be the foot of the tower.

### Step 3: Identify the Angles and Heights
- The height of the building (AB) = 12 m.
- The angle of elevation from A to P is \(60^\circ\).
- The angle of depression from A to Q is \(\theta\).

### Step 4: Use the Angle of Elevation
Using the angle of elevation, we can set up a right triangle \(APQ\):
- In triangle \(APQ\), we have:
\[ \tan(60^\circ) = \frac{h_{tower} - 12}{BQ} \]
where \(h_{tower}\) is the height of the tower, and \(BQ\) is the horizontal distance from the base of the building to the base of the tower.

Since \(\tan(60^\circ) = \sqrt{3}\):
\[ \sqrt{3} = \frac{h_{tower} - 12}{BQ} \quad \text{(1)} \]

### Step 5: Use the Angle of Depression
Using the angle of depression, we can set up another right triangle \(ABQ\):
- From the angle of depression, we know:
\[ \tan(\theta) = \frac{12}{BQ} \]
Given that \(\tan(\theta) = \frac{3}{4}\):
\[ \frac{3}{4} = \frac{12}{BQ} \quad \text{(2)} \]

### Step 6: Solve for BQ
From equation (2):
\[ BQ = \frac{12 \times 4}{3} = 16 \text{ m} \]

### Step 7: Substitute BQ into Equation (1)
Now substitute \(BQ\) back into equation (1):
\[ \sqrt{3} = \frac{h_{tower} - 12}{16} \]
Multiplying both sides by 16:
\[ 16\sqrt{3} = h_{tower} - 12 \]
Thus,
\[ h_{tower} = 16\sqrt{3} + 12 \]

### Step 8: Calculate the Height of the Tower
Now substituting \(\sqrt{3} = 1.73\):
\[ h_{tower} = 16 \times 1.73 + 12 \]
Calculating:
\[ h_{tower} = 27.68 + 12 = 39.68 \text{ m} \]

### Final Answer
The height of the tower is \(39.68\) m.

---