The angles of depression from the top of a light house of two boats are `45^@ and 30^@` towards the west. If the two boats are 5m apart, then the heighat of the light house is

Options

A. (2.5V3 -1)m
B. 2.5 (V3 -1)m
C. (2.5 V 3 + 1)m
D. 2.5 (V3 + 1) m

Answer

To solve the problem, we will use the concept of angles of depression and some basic trigonometry.

### Step-by-step Solution:

1. **Understanding the Problem:**
- We have a lighthouse from which the angles of depression to two boats are given as \(45^\circ\) and \(30^\circ\).
- The distance between the two boats is \(5\) meters.

2. **Setting Up the Diagram:**
- Let the height of the lighthouse be \(h\).
- Let the distance from the base of the lighthouse to the first boat (where the angle of depression is \(45^\circ\)) be \(d_1\).
- Let the distance from the base of the lighthouse to the second boat (where the angle of depression is \(30^\circ\)) be \(d_2\).
- According to the problem, the distance between the two boats is \(d_2 - d_1 = 5\) meters.

3. **Using Trigonometry:**
- For the first boat (angle of depression \(45^\circ\)):
\[ \tan(45^\circ) = \frac{h}{d_1} \]
Since \(\tan(45^\circ) = 1\), we have:
\[ h = d_1 \]

- For the second boat (angle of depression \(30^\circ\)):
\[ \tan(30^\circ) = \frac{h}{d_2} \]
Since \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\), we have:
\[ h = \frac{d_2}{\sqrt{3}} \]

4. **Setting Up the Equation:**
- From the two equations we derived:
\[ d_1 = h \]
\[ d_2 = h\sqrt{3} \]
- Now substituting into the distance equation:
\[ d_2 - d_1 = 5 \]
\[ h\sqrt{3} - h = 5 \]
\[ h(\sqrt{3} - 1) = 5 \]

5. **Solving for \(h\):**
- Now, we can solve for \(h\):
\[ h = \frac{5}{\sqrt{3} - 1} \]
- To simplify, multiply the numerator and the denominator by the conjugate of the denominator:
\[ h = \frac{5(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{5(\sqrt{3} + 1)}{3 - 1} = \frac{5(\sqrt{3} + 1)}{2} \]

6. **Final Calculation:**
- Thus, the height of the lighthouse is:
\[ h = \frac{5(\sqrt{3} + 1)}{2} \approx 5 \times 1.366 = 6.83 \text{ meters (approximately)} \]

### Final Answer:
The height of the lighthouse is approximately \(6.83\) meters.