A man in a boat rowing away from a light house 100 m high takes 2 minutes to change the angle of elevation of the top of the light house from 60° to 30°. Find the speed of the boat in meters per minute. [Use `sqrt(3) = 1.732`)

Answer

To solve the problem step by step, we can follow these calculations:

### Step 1: Understand the Problem
We have a lighthouse that is 100 m high. A man in a boat rows away from the lighthouse, changing the angle of elevation from 60° to 30° in 2 minutes. We need to find the speed of the boat in meters per minute.

### Step 2: Set Up the Diagram
Let:
- A be the top of the lighthouse.
- B be the base of the lighthouse.
- C be the position of the boat when the angle of elevation is 60°.
- D be the position of the boat when the angle of elevation is 30°.

### Step 3: Identify the Right Triangles
In triangle ABC (where angle ACB = 60°):
- AB = height of the lighthouse = 100 m
- CB = distance from the base of the lighthouse to the boat when the angle is 60°.

In triangle ABD (where angle ADB = 30°):
- AB = height of the lighthouse = 100 m
- DB = distance from the base of the lighthouse to the boat when the angle is 30°.

### Step 4: Use Trigonometric Ratios
For triangle ABC:
\[ \tan(60°) = \frac{AB}{CB} \implies \sqrt{3} = \frac{100}{CB} \implies CB = \frac{100}{\sqrt{3}} \text{ m} \]

For triangle ABD:
\[ \tan(30°) = \frac{AB}{DB} \implies \frac{1}{\sqrt{3}} = \frac{100}{DB} \implies DB = 100\sqrt{3} \text{ m} \]

### Step 5: Calculate the Distance the Boat Rowed
The distance CD that the boat has rowed can be calculated as:
\[ CD = DB - CB = 100\sqrt{3} - \frac{100}{\sqrt{3}} \]

### Step 6: Simplify the Expression
To simplify \(CD\):
1. Find a common denominator:
\[ CD = 100\sqrt{3} - \frac{100}{\sqrt{3}} = \frac{100\sqrt{3} \cdot \sqrt{3}}{\sqrt{3}} - \frac{100}{\sqrt{3}} = \frac{300 - 100}{\sqrt{3}} = \frac{200}{\sqrt{3}} \]

### Step 7: Find the Speed of the Boat
The distance CD is covered in 2 minutes. Let the speed of the boat be \(x\) meters per minute:
\[ 2x = \frac{200}{\sqrt{3}} \implies x = \frac{100}{\sqrt{3}} \text{ m/min} \]

### Step 8: Substitute the Value of \(\sqrt{3}\)
Using \(\sqrt{3} \approx 1.732\):
\[ x = \frac{100}{1.732} \approx 57.80 \text{ m/min} \]

### Final Answer
The speed of the boat is approximately **57.80 meters per minute**.

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