A man of height 6 ft. observes the top of a tower and the foot of the tower at angles of `45^@` and `30^@` of elevation and depression respectively. The height of the tower is

Options

A. 16.39 m
B. 14.59 m
C. 14.29 m
D. none of these

Answer

To find the height of the tower, we can follow these steps:

### Step 1: Understand the Problem
We have a man who is 6 feet tall. He observes the top of a tower at an angle of elevation of 45 degrees and the foot of the tower at an angle of depression of 30 degrees. We need to find the height of the tower.

### Step 2: Define Variables
Let:
- \( h \) = height of the tower (in feet)
- The height of the man = 6 feet
- The distance from the man to the base of the tower = \( x \)

### Step 3: Set Up the Right Triangles
1. **For the angle of elevation (45 degrees)**:
- The height from the man's eye level to the top of the tower is \( h - 6 \) (since the man is 6 feet tall).
- According to the tangent function:
\[ \tan(45^\circ) = \frac{h - 6}{x} \]
Since \( \tan(45^\circ) = 1 \):
\[ 1 = \frac{h - 6}{x} \implies h - 6 = x \quad \text{(Equation 1)} \]

2. **For the angle of depression (30 degrees)**:
- The height from the man's eye level down to the foot of the tower is 6 feet.
- According to the tangent function:
\[ \tan(30^\circ) = \frac{6}{x} \]
Since \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \):
\[ \frac{1}{\sqrt{3}} = \frac{6}{x} \implies x = 6\sqrt{3} \quad \text{(Equation 2)} \]

### Step 4: Substitute Equation 2 into Equation 1
From Equation 1, we have:
\[ h - 6 = x \]
Substituting \( x = 6\sqrt{3} \):
\[ h - 6 = 6\sqrt{3} \]
Adding 6 to both sides:
\[ h = 6\sqrt{3} + 6 \]

### Step 5: Calculate the Height of the Tower
Now we can calculate \( h \):
- First, we need to find the value of \( 6\sqrt{3} \):
\[ \sqrt{3} \approx 1.732 \implies 6\sqrt{3} \approx 6 \times 1.732 \approx 10.392 \]
Thus,
\[ h \approx 10.392 + 6 \approx 16.392 \text{ feet} \]

### Final Answer
The height of the tower is approximately **16.39 feet**.

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