If from point 100 m above the ground the angles of depression of two objects due south on the ground are `60^(@) and 45^(@)`, then the distance between the objects is :

Options

A. `(50(3-sqrt3))/(3)m`
B. `(50(3+sqrt3))/(3)m`
C. `(100(3+sqrt3))/(3)m`
D. `(100(3-sqrt3))/(3)m`

Answer

To solve the problem, we need to find the distance between two objects on the ground based on the angles of depression from a point 100 m above the ground.

### Step-by-Step Solution:

1. **Understanding the Scenario**:
- We have a point (let's call it A) that is 100 m above the ground.
- From point A, the angles of depression to two objects on the ground are 60° and 45°.

2. **Drawing the Diagram**:
- Draw a vertical line representing the height of 100 m (point A).
- Draw horizontal lines from point A to the two objects on the ground (let's call them O1 and O2).
- The angle of depression to O1 is 60° and to O2 is 45°.

3. **Identifying the Triangles**:
- From point A to O1, we can form a right triangle (A-O1) where:
- The height (opposite side) = 100 m
- The angle of depression = 60°
- From point A to O2, we can form another right triangle (A-O2) where:
- The height (opposite side) = 100 m
- The angle of depression = 45°

4. **Calculating Distances**:
- For triangle A-O1 (with angle 60°):
- Using the tangent function:
\[ \tan(60°) = \frac{\text{opposite}}{\text{adjacent}} = \frac{100}{x_2} \]
- Rearranging gives:
\[ x_2 = \frac{100}{\tan(60°)} = \frac{100}{\sqrt{3}} \approx 100 \cdot \frac{1}{\sqrt{3}} = 100\sqrt{3} \text{ m} \]

- For triangle A-O2 (with angle 45°):
- Using the tangent function:
\[ \tan(45°) = \frac{\text{opposite}}{\text{adjacent}} = \frac{100}{x_1} \]
- Rearranging gives:
\[ x_1 = \frac{100}{\tan(45°)} = 100 \text{ m} \]

5. **Finding the Distance Between the Objects**:
- The distance between the two objects O1 and O2 (let's call it d) is:
\[ d = x_1 - x_2 = 100 - 100\sqrt{3} \]
- Simplifying gives:
\[ d = 100(1 - \sqrt{3}) \text{ m} \]

6. **Final Answer**:
- The distance between the two objects is:
\[ d = 100(1 - \sqrt{3}) \text{ m} \]