From the top of a light house, it is observed that a ship is sailing directly towards it and the angle of depression of the ship changes from `30^(@) " to " 45^(@)` in 10 minutes. Assuming that the ship is sailing with uniform speed, calculate in how much more time (in minutes) will the ship reach the light house?

Answer

To solve the problem, we need to analyze the situation involving the lighthouse and the ship using trigonometry. Here’s a step-by-step solution:

### Step 1: Understand the Geometry
Let’s denote:
- The height of the lighthouse as \( h \).
- The distance of the ship from the base of the lighthouse when the angle of depression is \( 30^\circ \) as \( d_1 \).
- The distance of the ship from the base of the lighthouse when the angle of depression is \( 45^\circ \) as \( d_2 \).

### Step 2: Set Up the Trigonometric Relationships
From the top of the lighthouse:
1. When the angle of depression is \( 30^\circ \):
\[ \tan(30^\circ) = \frac{h}{d_1} \implies d_1 = h \cdot \sqrt{3} \]
(since \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \))

2. When the angle of depression is \( 45^\circ \):
\[ \tan(45^\circ) = \frac{h}{d_2} \implies d_2 = h \]
(since \( \tan(45^\circ) = 1 \))

### Step 3: Calculate the Distance Traveled by the Ship
The ship travels from \( d_1 \) to \( d_2 \) in 10 minutes. The distance traveled by the ship in this time is:
\[ d_1 - d_2 = h\sqrt{3} - h = h(\sqrt{3} - 1) \]

Let the speed of the ship be \( v \). The distance traveled in 10 minutes can also be expressed as:
\[ 10v \]

### Step 4: Set Up the Equation
Equating the two expressions for the distance traveled:
\[ h(\sqrt{3} - 1) = 10v \implies v = \frac{h(\sqrt{3} - 1)}{10} \]

### Step 5: Calculate the Remaining Distance to the Lighthouse
The remaining distance for the ship to reach the lighthouse from \( d_2 \) is simply \( d_2 \):
\[ d_2 = h \]

### Step 6: Calculate the Time to Reach the Lighthouse
The time \( t \) taken to cover the remaining distance \( h \) at speed \( v \) is given by:
\[ t = \frac{h}{v} = \frac{h}{\frac{h(\sqrt{3} - 1)}{10}} = \frac{10}{\sqrt{3} - 1} \]

### Step 7: Calculate the Total Time
To find the total time in minutes for the ship to reach the lighthouse, we need to convert the time calculated above:
\[ t = \frac{10}{\sqrt{3} - 1} \]
To simplify this expression, we can multiply the numerator and denominator by \( \sqrt{3} + 1 \):
\[ t = \frac{10(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{10(\sqrt{3} + 1)}{3 - 1} = 5(\sqrt{3} + 1) \]

### Step 8: Calculate the Numerical Value
Using \( \sqrt{3} \approx 1.732 \):
\[ t \approx 5(1.732 + 1) = 5(2.732) \approx 13.66 \text{ minutes} \]

### Step 9: Find the Additional Time
Since the ship has already traveled for 10 minutes, the additional time required to reach the lighthouse is:
\[ t - 10 \approx 13.66 - 10 = 3.66 \text{ minutes} \]

### Final Answer
The ship will reach the lighthouse in approximately **3.66 minutes** more.