As observed from the top of a 80 m tall lighthouse, the angle of depression of two ships, on the same side of the light house in horizontal line with its base, are `30^(@) and 40^(@)` respectively . Find the distance between the two ships. Given your answer correct to the nearest metre

Answer

To solve the problem step by step, we will use trigonometric principles to find the distances from the lighthouse to each ship and then calculate the distance between the two ships.

### Step 1: Draw the Diagram
Draw a vertical line representing the lighthouse of height 80 m. Label the top of the lighthouse as point A and the base as point B. The two ships will be represented as points C and D on the horizontal line from point B.

### Step 2: Identify Angles and Distances
From the top of the lighthouse (point A), the angle of depression to ship C is \(40^\circ\) and to ship D is \(30^\circ\). The height of the lighthouse (AB) is 80 m.

### Step 3: Use Trigonometry for Ship D (Angle 30°)
In triangle ABD:
- The angle of depression to ship D is \(30^\circ\).
- Using the tangent function:
\[ \tan(30^\circ) = \frac{AB}{BD} \]
Where \(AB = 80\) m and \(BD\) is the distance from the base of the lighthouse to ship D.

Since \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\):
\[ \frac{1}{\sqrt{3}} = \frac{80}{BD} \]
Rearranging gives:
\[ BD = 80 \sqrt{3} \]
Calculating \(BD\):
\[ BD \approx 80 \times 1.732 = 138.56 \text{ m} \]

### Step 4: Use Trigonometry for Ship C (Angle 40°)
In triangle ABC:
- The angle of depression to ship C is \(40^\circ\).
- Using the tangent function:
\[ \tan(40^\circ) = \frac{AB}{BC} \]
Where \(BC\) is the distance from the base of the lighthouse to ship C.

Since \(\tan(40^\circ) \approx 0.8390\):
\[ 0.8390 = \frac{80}{BC} \]
Rearranging gives:
\[ BC = \frac{80}{0.8390} \]
Calculating \(BC\):
\[ BC \approx 95.352 \text{ m} \]

### Step 5: Calculate the Distance Between the Two Ships
The distance between the two ships (CD) is given by:
\[ CD = BD - BC \]
Substituting the values we found:
\[ CD = 138.56 - 95.352 \approx 43.208 \text{ m} \]

### Step 6: Round to the Nearest Meter
Rounding \(43.208\) to the nearest meter gives:
\[ \text{Distance between the two ships} \approx 43 \text{ m} \]

### Final Answer
The distance between the two ships is **43 meters**.

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