From the top of a cliff, the angle of depression of the top and bottom of a tower are observed to be `45^(@)` and `60^(@)` respectively. If the height of the tower is 20 m.
Find :
(i) the height of the cliff
(ii) the distance between the cliff and the tower.

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Problem
We have a cliff and a tower. The height of the tower (AB) is given as 20 m. The angles of depression from the top of the cliff (C) to the top (A) and bottom (B) of the tower are 45° and 60° respectively. We need to find the height of the cliff (CA) and the distance between the cliff and the tower (BC).

### Step 2: Draw a Diagram
Let's label the points:
- C: Top of the cliff
- A: Top of the tower
- B: Bottom of the tower
- D: Point directly below C on the ground level

From the problem, we know:
- Height of the tower (AB) = 20 m
- Angle of depression to A = 45°
- Angle of depression to B = 60°

### Step 3: Set Up the Triangles
Using the angles of depression, we can set up two right triangles:
1. Triangle CAD (for angle 45°)
2. Triangle CBD (for angle 60°)

### Step 4: Calculate the Distance BC (x)
In triangle CAD:
- The angle of depression to A is 45°, which means angle CAD = 45°.
- Using the tangent function:
\[ \tan(45°) = \frac{CA}{AD} \]
Since \(\tan(45°) = 1\):
\[ CA = AD \implies AD = h \]
where \(h\) is the height of the cliff.

### Step 5: Calculate the Height of the Cliff (h)
In triangle CBD:
- The angle of depression to B is 60°, which means angle CBD = 60°.
- Using the tangent function:
\[ \tan(60°) = \frac{CA + AB}{BC} \]
Substituting the known values:
\[ \sqrt{3} = \frac{h + 20}{x} \]
Since \(x = h\):
\[ \sqrt{3} = \frac{h + 20}{h} \]
Rearranging gives:
\[ \sqrt{3}h = h + 20 \]
\[ h(\sqrt{3} - 1) = 20 \]
\[ h = \frac{20}{\sqrt{3} - 1} \]

### Step 6: Rationalize the Denominator
To rationalize:
\[ h = \frac{20(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{20(\sqrt{3} + 1)}{3 - 1} = 10(\sqrt{3} + 1) \]

### Step 7: Find the Total Height of the Cliff
The total height of the cliff (CA) is:
\[ CA = h + AB = 10(\sqrt{3} + 1) + 20 \]
Calculating this:
\[ CA = 10\sqrt{3} + 10 + 20 = 10\sqrt{3} + 30 \]

### Step 8: Calculate the Distance BC
Since \(BC = h\):
\[ BC = 10(\sqrt{3} + 1) \]

### Final Results
1. Height of the cliff (CA) = \(10\sqrt{3} + 30\) meters
2. Distance between the cliff and the tower (BC) = \(10(\sqrt{3} + 1)\) meters