A 1.6 m tall observer is 45 metres away from a tower .The angle of elevation from his eye to the top of the tower is `30^(@)` , then the height of the tower in metres is (Take `sqrt(3)=1.732)`

Options

A. `25 . 98`
B. `26 . 58`
C. `27 . 58`
D. `27 . 98`

Answer

To solve the problem step by step, we will use trigonometry to find the height of the tower.

### Step 1: Understand the problem
We have an observer who is 1.6 meters tall and is standing 45 meters away from a tower. The angle of elevation from the observer's eyes to the top of the tower is 30 degrees. We need to find the total height of the tower.

### Step 2: Draw a diagram
Let's visualize the situation:
- Let point A be the top of the tower.
- Let point B be the observer's eyes (which is 1.6 meters above the ground).
- Let point C be the base of the tower.
- The distance from B to C (the horizontal distance) is 45 meters.
- The angle of elevation from B to A is 30 degrees.

### Step 3: Use trigonometry to find the height from B to A
We can use the tangent function, which is defined as the ratio of the opposite side to the adjacent side in a right triangle. In our case:
- The opposite side is the height from the observer's eyes (B) to the top of the tower (A), which we can denote as \( h \).
- The adjacent side is the distance from the observer to the tower (BC), which is 45 meters.

Using the tangent of the angle of elevation:
\[ \tan(30^\circ) = \frac{h}{45} \]

### Step 4: Substitute the value of \( \tan(30^\circ) \)
We know that:
\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \]
So we can substitute this into our equation:
\[ \frac{1}{\sqrt{3}} = \frac{h}{45} \]

### Step 5: Solve for \( h \)
Cross-multiplying gives us:
\[ h = 45 \cdot \frac{1}{\sqrt{3}} = \frac{45}{\sqrt{3}} \]

### Step 6: Rationalize the denominator
To simplify \( \frac{45}{\sqrt{3}} \), we can multiply the numerator and denominator by \( \sqrt{3} \):
\[ h = \frac{45 \sqrt{3}}{3} = 15 \sqrt{3} \]

### Step 7: Substitute the value of \( \sqrt{3} \)
Given \( \sqrt{3} \approx 1.732 \):
\[ h = 15 \cdot 1.732 = 25.98 \text{ meters} \]

### Step 8: Add the height of the observer
Now, we need to add the height of the observer (1.6 meters) to the height \( h \):
\[ \text{Total height of the tower} = h + 1.6 = 25.98 + 1.6 = 27.58 \text{ meters} \]

### Final Answer
The height of the tower is approximately **27.58 meters**.

---