From a boat, which is moving towards a bridge, the angle of elevation of bridge is `30^(@)`. After 10 minutes, the angle of elevation becomes `60^(@)`. Find how much more time will take the boat to reach at the bridge?

Answer

To solve the problem step by step, we will use trigonometric ratios and the information provided in the question.

### Step 1: Understand the Problem
We have a boat moving towards a bridge. The angle of elevation of the bridge from the boat changes from \(30^\circ\) to \(60^\circ\) after 10 minutes. We need to find out how much more time it will take for the boat to reach the bridge.

### Step 2: Set Up the Diagram
Let:
- \(AB\) be the height of the bridge.
- \(D\) be the initial position of the boat when the angle of elevation is \(30^\circ\).
- \(C\) be the position of the boat after 10 minutes when the angle of elevation is \(60^\circ\).
- \(B\) be the point directly below the bridge.

### Step 3: Use Trigonometric Ratios
From triangle \(ADB\) (where \(D\) is the position of the boat when the angle is \(30^\circ\)):
\[ \tan(30^\circ) = \frac{AB}{BD} \]
We know that \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\). Therefore,
\[ \frac{1}{\sqrt{3}} = \frac{H}{BD} \implies BD = H \sqrt{3} \]

From triangle \(ACB\) (where \(C\) is the position of the boat after 10 minutes when the angle is \(60^\circ\)):
\[ \tan(60^\circ) = \frac{AB}{BC} \]
We know that \(\tan(60^\circ) = \sqrt{3}\). Therefore,
\[ \sqrt{3} = \frac{H}{BC} \implies BC = \frac{H}{\sqrt{3}} \]

### Step 4: Calculate the Distance Covered by the Boat
The distance the boat has traveled in 10 minutes is \(DC\):
\[ DC = BD - BC = H\sqrt{3} - \frac{H}{\sqrt{3}} \]
To simplify this, we find a common denominator:
\[ DC = H\sqrt{3} - \frac{H}{\sqrt{3}} = \frac{3H - H}{\sqrt{3}} = \frac{2H}{\sqrt{3}} \]

### Step 5: Calculate the Speed of the Boat
The speed of the boat can be calculated as:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{DC}{10 \text{ minutes}} = \frac{\frac{2H}{\sqrt{3}}}{10} = \frac{2H}{10\sqrt{3}} = \frac{H}{5\sqrt{3}} \text{ (units of H per minute)} \]

### Step 6: Calculate the Time to Reach the Bridge from Point C
Now we need to calculate the time taken to cover the distance \(BC\):
\[ BC = \frac{H}{\sqrt{3}} \]
Using the formula for time:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{BC}{\text{Speed}} = \frac{\frac{H}{\sqrt{3}}}{\frac{H}{5\sqrt{3}}} \]
This simplifies to:
\[ \text{Time} = \frac{H}{\sqrt{3}} \times \frac{5\sqrt{3}}{H} = 5 \text{ minutes} \]

### Conclusion
The boat will take an additional 5 minutes to reach the bridge after the 10 minutes have passed.

### Final Answer
The boat will take **5 more minutes** to reach the bridge.

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