A guard observes an enemy boat, from an observation tower at a height of 180 m above sea level, to be at an angle of depression of `29^(@)`
After some time, it is observed that the boat is 200 m from the foot of the observation tower. Calculate the new angle of depression.

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Understand the Problem
We have a guard at a height of 180 meters observing a boat at an angle of depression of 29 degrees. After some time, the boat is 200 meters away from the foot of the tower. We need to find the new angle of depression.

**Hint:** Visualize the scenario by drawing a diagram with the tower, the guard, the boat, and the angles involved.

### Step 2: Draw the Diagram
Draw a vertical line representing the observation tower with a height of 180 m. Mark the top of the tower as point A (where the guard is) and the foot of the tower as point B. Draw a horizontal line from point A to represent the line of sight parallel to the ground. Mark the position of the boat as point C.

**Hint:** Label the points clearly: A (top of the tower), B (foot of the tower), and C (boat's position).

### Step 3: Identify the Angles
The angle of depression from point A to point C is given as 29 degrees. The new position of the boat (point C) is now 200 meters from point B.

**Hint:** Remember that the angle of depression is equal to the angle of elevation from the boat to the guard.

### Step 4: Set Up the Right Triangle
In triangle ABC, the height (AB) is 180 m (the height of the tower), and the distance (BC) is 200 m (the distance from the boat to the foot of the tower). We need to find the new angle of depression, which we will denote as θ.

**Hint:** Use the properties of right triangles to relate the sides and angles.

### Step 5: Use the Tangent Function
The tangent of the angle θ can be expressed as:
\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{AB}{BC} \]
Substituting the known values:
\[ \tan(\theta) = \frac{180}{200} \]

**Hint:** Simplify the fraction before calculating the angle.

### Step 6: Simplify the Expression
\[ \tan(\theta) = \frac{180}{200} = \frac{9}{10} \]

**Hint:** This fraction can be directly used to find the angle using the inverse tangent function.

### Step 7: Calculate the Angle
Now, use the inverse tangent function to find θ:
\[ \theta = \tan^{-1}\left(\frac{9}{10}\right) \]

**Hint:** Use a scientific calculator or trigonometric tables to find the angle.

### Step 8: Find the Final Angle
Calculating this gives:
\[ \theta \approx 41.99^\circ \]
This can be converted to degrees and minutes:
\[ \theta \approx 41^\circ 59' \]

**Hint:** Remember to convert decimal degrees to degrees and minutes if necessary.

### Final Answer
The new angle of depression is approximately \( 41^\circ 59' \).

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