From the Masthead of a ship, the angle of Depression of boat is `60^@`, If the mast head is 150 m long , then the distance of the boat from the ship is :

Options

A. 86.6 m
B. 68.6 m
C. 66.8 m
D. none of these

Answer

To solve the problem, we need to find the distance of the boat from the ship using the given information. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We have a ship with a mast that is 150 meters tall. From the top of the mast, the angle of depression to the boat is 60 degrees. We need to find the horizontal distance from the base of the mast (the ship) to the boat.

### Step 2: Draw a Diagram
1. Draw a vertical line representing the mast of the ship (AB) which is 150 meters tall.
2. Draw a horizontal line from the base of the mast (point A) to the position of the boat (point C).
3. Connect the top of the mast (point B) to the boat (point C).
4. Mark the angle of depression from point B to point C as 60 degrees.

### Step 3: Identify the Triangle
In the triangle ABC:
- AB is the height of the mast (150 m).
- BC is the distance from the ship to the boat (which we need to find).
- Angle ACB is the angle of depression, which is equal to 60 degrees.

### Step 4: Use Trigonometric Ratios
We can use the tangent function, which relates the opposite side (AB) to the adjacent side (BC):
\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \]
Here, \(\theta = 60^\circ\), Opposite = AB = 150 m, and Adjacent = BC.

### Step 5: Set Up the Equation
Using the tangent function:
\[ \tan(60^\circ) = \frac{AB}{BC} \]
Substituting the known values:
\[ \sqrt{3} = \frac{150}{BC} \]

### Step 6: Solve for BC
Rearranging the equation gives:
\[ BC = \frac{150}{\sqrt{3}} \]

### Step 7: Rationalize the Denominator
To simplify \(\frac{150}{\sqrt{3}}\), we multiply the numerator and the denominator by \(\sqrt{3}\):
\[ BC = \frac{150 \cdot \sqrt{3}}{3} = 50\sqrt{3} \]

### Step 8: Calculate the Approximate Value
Now, we can calculate the approximate value of \(BC\):
\[ BC \approx 50 \cdot 1.732 \approx 86.6 \text{ meters} \]

### Final Answer
The distance of the boat from the ship is approximately **86.6 meters**.

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