A man on the top of a cliff 100 meter high, observes the angles of depression of two points on the opposite sides of the cliff as `30^(@)` and `60^(@)` respectively. Find the difference between the two points.

Answer

To solve the problem step by step, we will use the concepts of trigonometry, specifically the tangent function, to find the distances from the cliff to the two points observed by the man.

### Step 1: Understand the problem
We have a cliff that is 100 meters high. A man at the top of the cliff observes two points on the ground at angles of depression of 30° and 60°. We need to find the horizontal distances from the base of the cliff to these two points and then calculate the difference between these distances.

### Step 2: Draw a diagram
Draw a vertical line representing the cliff and label its height as 100 meters. From the top of the cliff, draw two lines downwards at angles of 30° and 60° to represent the lines of sight to the two points on the ground. Label the horizontal distances from the base of the cliff to the two points as x (for the point at 30°) and y (for the point at 60°).

### Step 3: Use the tangent function
For the angle of depression, we can use the tangent function, which relates the opposite side (height of the cliff) to the adjacent side (horizontal distance).

1. For the angle of depression of 30°:
\[ \tan(30°) = \frac{\text{Height of cliff}}{x} = \frac{100}{x} \]
We know that \(\tan(30°) = \frac{1}{\sqrt{3}}\). Therefore:
\[ \frac{1}{\sqrt{3}} = \frac{100}{x} \]
Rearranging gives:
\[ x = 100 \sqrt{3} \]

2. For the angle of depression of 60°:
\[ \tan(60°) = \frac{\text{Height of cliff}}{y} = \frac{100}{y} \]
We know that \(\tan(60°) = \sqrt{3}\). Therefore:
\[ \sqrt{3} = \frac{100}{y} \]
Rearranging gives:
\[ y = \frac{100}{\sqrt{3}} \]

### Step 4: Find the difference between the two points
Now that we have both distances, we can find the difference between them:
\[ \text{Difference} = x - y = 100 \sqrt{3} - \frac{100}{\sqrt{3}} \]

### Step 5: Simplify the expression
To simplify the expression, we can find a common denominator:
\[ \text{Difference} = 100 \sqrt{3} - \frac{100}{\sqrt{3}} = \frac{100 \cdot 3}{\sqrt{3}} - \frac{100}{\sqrt{3}} = \frac{300 - 100}{\sqrt{3}} = \frac{200}{\sqrt{3}} \]

### Final Answer
Thus, the difference between the two points is:
\[ \frac{200}{\sqrt{3}} \text{ meters} \]