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 about the angles of elevation.

### Step 1: Understanding the Setup
Let:
- \( A \) be the position of the boat when the angle of elevation is \( 30^\circ \).
- \( B \) be the position of the bridge.
- \( C \) be the position of the boat after 10 minutes when the angle of elevation is \( 60^\circ \).

### Step 2: Drawing the Right Triangles
From point \( A \) (initial position of the boat):
- The angle of elevation to the bridge \( B \) is \( 30^\circ \).
- Let the height of the bridge \( B \) above the water be \( h \).
- The horizontal distance from \( A \) to the base of the bridge \( D \) is \( AD \).

From point \( C \) (position of the boat after 10 minutes):
- The angle of elevation to the bridge \( B \) is \( 60^\circ \).
- The horizontal distance from \( C \) to the base of the bridge \( D \) is \( CD \).

### Step 3: Applying Trigonometric Ratios
Using the tangent function, we can set up the following equations:

1. From triangle \( ABD \):
\[ \tan(30^\circ) = \frac{h}{AD} \]
We know that \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \), so:
\[ \frac{1}{\sqrt{3}} = \frac{h}{AD} \implies h = \frac{AD}{\sqrt{3}} \quad \text{(1)} \]

2. From triangle \( CBD \):
\[ \tan(60^\circ) = \frac{h}{CD} \]
We know that \( \tan(60^\circ) = \sqrt{3} \), so:
\[ \sqrt{3} = \frac{h}{CD} \implies h = CD \cdot \sqrt{3} \quad \text{(2)} \]

### Step 4: Equating the Heights
From equations (1) and (2):
\[ \frac{AD}{\sqrt{3}} = CD \cdot \sqrt{3} \]
Cross-multiplying gives:
\[ AD = 3CD \quad \text{(3)} \]

### Step 5: Finding Distances
The distance \( AC \) traveled by the boat in 10 minutes is:
\[ AC = AD - CD \]
Substituting \( AD \) from equation (3):
\[ AC = 3CD - CD = 2CD \]

### Step 6: Speed of the Boat
The speed of the boat can be calculated as:
\[ \text{Speed} = \frac{AC}{10 \text{ minutes}} = \frac{2CD}{10} = \frac{CD}{5} \quad \text{(4)} \]

### Step 7: Remaining Distance to Cover
The remaining distance \( CD \) is the distance from the position of the boat \( C \) to the bridge \( B \).

### Step 8: Time to Reach the Bridge
Using the speed from equation (4):
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{CD}{\frac{CD}{5}} = 5 \text{ minutes} \]

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

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