A man on the deck of a ship 12 m above water level, observes that the angle of elevation of the top of a cliff is `60^(@)` and the angle of depression of the base of the is `30^(@)`. Find the distance of the cliff from the ship and the height of the cliff. [use `sqrt(3)=1.732`)

Answer

To solve the problem step by step, we will use trigonometric ratios to find the distance of the cliff from the ship and the height of the cliff.

### Given:
- Height of the ship above water level (AB) = 12 m
- Angle of elevation to the top of the cliff (C) = 60°
- Angle of depression to the base of the cliff (D) = 30°

### Step 1: Draw a diagram
1. Draw a horizontal line representing the water level.
2. Mark point A as the position of the ship, 12 m above the water level.
3. Mark point D as the base of the cliff and point C as the top of the cliff.
4. Draw vertical lines from points C and D to the water level, and connect points A, C, and D.

### Step 2: Find the distance AE (horizontal distance from the ship to the base of the cliff)
Using the angle of depression (30°) from point A to point D:

In triangle AED:
- AE = horizontal distance from the ship to the base of the cliff
- AD = vertical distance from the ship to the base of the cliff (which is the height of the ship, AB = 12 m)

Using the tangent function:
\[ \tan(30°) = \frac{ED}{AE} \]
Where ED = height of the ship = 12 m.

From trigonometric values:
\[ \tan(30°) = \frac{1}{\sqrt{3}} \]
Thus,
\[ \frac{1}{\sqrt{3}} = \frac{12}{AE} \]

Cross-multiplying gives:
\[ AE = 12 \cdot \sqrt{3} \]

### Step 3: Find the height of the cliff (CD)
Using the angle of elevation (60°) from point A to point C:

In triangle AEC:
- CE = height of the cliff above the ship (CD)
- AE = horizontal distance from the ship to the base of the cliff (found in Step 2)

Using the tangent function:
\[ \tan(60°) = \frac{CE}{AE} \]
From trigonometric values:
\[ \tan(60°) = \sqrt{3} \]
Thus,
\[ \sqrt{3} = \frac{CE}{12 \cdot \sqrt{3}} \]

Cross-multiplying gives:
\[ CE = 12 \cdot \sqrt{3} \cdot \sqrt{3} = 12 \cdot 3 = 36 m \]

### Step 4: Find the total height of the cliff (CD)
The total height of the cliff (CD) is the sum of the height of the ship (AB) and the height of the cliff above the ship (CE):
\[ CD = AB + CE = 12 + 36 = 48 m \]

### Final Results:
- Distance of the cliff from the ship (AE) = \(12 \sqrt{3}\) m
- Height of the cliff (CD) = 48 m