The angles of depression of two ships from the top of a light house are `45^(@)` and `30^(@)` towards east. If the ships are 200 m apart, find the height of light house.

Answer

To solve the problem, we need to find the height of the lighthouse based on the angles of depression to two ships that are 200 meters apart. Let's denote the height of the lighthouse as \( h \).

### Step 1: Understanding the Angles of Depression
The angle of depression from the top of the lighthouse to the ships is given as \( 45^\circ \) for the first ship (let's call it Ship A) and \( 30^\circ \) for the second ship (let's call it Ship B).

### Step 2: Drawing the Diagram
We can visualize the situation by drawing a right triangle for each ship. The height of the lighthouse is the vertical side of the triangle, and the horizontal distance from the base of the lighthouse to each ship is the horizontal side.

### Step 3: Setting Up the Right Triangles
1. For Ship A (angle of depression \( 45^\circ \)):
- Using the tangent function:
\[ \tan(45^\circ) = \frac{h}{d_1} \]
Since \( \tan(45^\circ) = 1 \):
\[ h = d_1 \]

2. For Ship B (angle of depression \( 30^\circ \)):
- Using the tangent function:
\[ \tan(30^\circ) = \frac{h}{d_2} \]
Since \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \):
\[ \frac{1}{\sqrt{3}} = \frac{h}{d_2} \implies d_2 = h \sqrt{3} \]

### Step 4: Relating the Distances
The distance between the two ships is given as 200 meters. Therefore, we can express this relationship as:
\[ d_2 - d_1 = 200 \]
Substituting the expressions for \( d_1 \) and \( d_2 \):
\[ h \sqrt{3} - h = 200 \]

### Step 5: Simplifying the Equation
Factoring out \( h \):
\[ h(\sqrt{3} - 1) = 200 \]

### Step 6: Solving for \( h \)
Now, we can solve for \( h \):
\[ h = \frac{200}{\sqrt{3} - 1} \]

### Step 7: Rationalizing the Denominator
To simplify further, we can rationalize the denominator:
\[ h = \frac{200(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{200(\sqrt{3} + 1)}{3 - 1} = \frac{200(\sqrt{3} + 1)}{2} = 100(\sqrt{3} + 1) \]

### Step 8: Final Calculation
Now we can calculate the approximate value of \( h \):
\[ h \approx 100(1.732 + 1) \approx 100(2.732) \approx 273.2 \text{ meters} \]

Thus, the height of the lighthouse is approximately **273.2 meters**.

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