A man whose eye leave is `1.5` meters above the ground observes the angle of elevation of the tower to be `60^(@).` If the distance of the man from the tower be 10 meters, the height of the tower is

Options

A. `(1.5 + 10 sqrt3)`
B. `10 sqrt3m`
C. `(1.5 + (10)/(sqrt3))m`
D. None of these

Answer

To find the height of the tower based on the given information, we can follow these steps:

### Step 1: Understand the Problem
We have a man standing at a distance of 10 meters from a tower. The height of his eyes from the ground is 1.5 meters. He observes the angle of elevation to the top of the tower to be 60 degrees.

### Step 2: Set Up the Diagram
Let's denote:
- Point A: The top of the tower
- Point B: The eye level of the man (1.5 meters above ground)
- Point C: The base of the tower
- Point D: The position of the man

The horizontal distance from the man (Point D) to the base of the tower (Point C) is 10 meters.

### Step 3: Identify the Height of the Tower
Let the height of the tower be \( h \). The height from the man's eye level to the top of the tower is \( h - 1.5 \) meters.

### Step 4: Use Trigonometry
In triangle ABC, we can use the tangent function since we have the angle of elevation and the opposite and adjacent sides:
\[ \tan(60^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{h - 1.5}{10} \]

### Step 5: Substitute the Value of \( \tan(60^\circ) \)
We know that:
\[ \tan(60^\circ) = \sqrt{3} \]
So, we can write:
\[ \sqrt{3} = \frac{h - 1.5}{10} \]

### Step 6: Solve for \( h - 1.5 \)
Multiplying both sides by 10:
\[ 10\sqrt{3} = h - 1.5 \]

### Step 7: Solve for \( h \)
Now, add 1.5 to both sides:
\[ h = 10\sqrt{3} + 1.5 \]

### Step 8: Calculate the Height
Now we can calculate the numerical value of \( h \):
Using \( \sqrt{3} \approx 1.732 \):
\[ h \approx 10 \times 1.732 + 1.5 = 17.32 + 1.5 = 18.82 \text{ meters} \]

### Final Answer
The height of the tower is approximately \( 18.82 \) meters.

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