A man is standing on the deck of a ship, which is 8m above water level. He observes the angle of elevation of the top of a hill as `60^@` and the angle of depression of the base of the hill as `30^@`. Calculate the distance of the hill from the ship and the height of the hill

Answer

To solve the problem, we will break it down into steps and use trigonometric ratios to find the distance of the hill from the ship and the height of the hill.

### Step 1: Understand the setup
- A man is standing on a ship's deck which is 8 meters above the water level.
- The angle of elevation to the top of the hill is \(60^\circ\).
- The angle of depression to the base of the hill is \(30^\circ\).

### Step 2: Draw the diagram
- Let \(C\) be the position of the man on the ship.
- Let \(E\) be the point directly below \(C\) on the water level, which is 8 meters above the water.
- Let \(A\) be the top of the hill and \(B\) be the base of the hill.
- The distance from \(C\) to \(B\) is \(d\) (the horizontal distance from the ship to the base of the hill).
- The height from \(E\) to \(A\) is \(h\) (the height of the hill above the water level).

### Step 3: Use trigonometric ratios
1. **For the angle of elevation to the top of the hill (\(60^\circ\)):**
- In triangle \(ACE\):
\[ \tan(60^\circ) = \frac{h}{d} \]
- Since \(\tan(60^\circ) = \sqrt{3}\), we have:
\[ \sqrt{3} = \frac{h}{d} \quad \Rightarrow \quad h = \sqrt{3} d \]

2. **For the angle of depression to the base of the hill (\(30^\circ\)):**
- In triangle \(CEB\):
\[ \tan(30^\circ) = \frac{8}{d} \]
- Since \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\), we have:
\[ \frac{1}{\sqrt{3}} = \frac{8}{d} \quad \Rightarrow \quad d = 8\sqrt{3} \]

### Step 4: Substitute to find height
- Substitute \(d = 8\sqrt{3}\) into the equation for \(h\):
\[ h = \sqrt{3} \cdot (8\sqrt{3}) = 8 \cdot 3 = 24 \text{ meters} \]

### Step 5: Calculate the total height of the hill
- The total height of the hill from the water level is:
\[ \text{Total height} = h + 8 = 24 + 8 = 32 \text{ meters} \]

### Step 6: Final answers
- The distance of the hill from the ship is \(d = 8\sqrt{3}\) meters.
- The height of the hill is \(32\) meters.

### Summary of Results
- Distance from the ship to the hill: \(8\sqrt{3} \approx 13.86\) meters
- Height of the hill: \(32\) meters