The angle of depression of a point on the horizontal from the top of a hill is `60^@`. If one has to walk 500 m to reach the top from this point, then the distance of this point from the base of the hill is

Options

A. `500 sqrt3` m
B. 250 m
C. `250 sqrt3m`
D. `250/sqrt3` m

Answer

To solve the problem, we will use the concept of trigonometry, specifically the sine function, to find the distance of the point from the base of the hill.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have a hill and a point on the horizontal ground from which the angle of depression to the top of the hill is given as \(60^\circ\).
- The distance walked to reach the top of the hill from this point is \(500\) meters.

2. **Identifying the Triangle**:
- Let's denote:
- \(A\): the top of the hill,
- \(B\): the point on the ground,
- \(M\): the foot of the hill (the base).
- The angle of depression from point \(A\) to point \(B\) is \(60^\circ\).
- The angle \(ABM\) (the angle at point \(B\)) is \(30^\circ\) because the angles of depression and elevation are equal.

3. **Using the Right Triangle**:
- In triangle \(ABM\):
- \(AB\) is the hypotenuse (which is \(500\) m),
- \(AM\) is the opposite side to angle \(30^\circ\),
- \(BM\) is the adjacent side to angle \(30^\circ\).

4. **Applying the Sine Function**:
- We can use the sine function to relate the opposite side to the hypotenuse:
\[ \sin(30^\circ) = \frac{AM}{AB} \]
- We know that \(\sin(30^\circ) = \frac{1}{2}\) and \(AB = 500\) m.

5. **Setting Up the Equation**:
- Plugging in the values we have:
\[ \frac{1}{2} = \frac{AM}{500} \]

6. **Solving for \(AM\)**:
- Rearranging the equation gives:
\[ AM = 500 \times \frac{1}{2} = 250 \text{ m} \]

7. **Finding Distance from the Base**:
- The distance from the base of the hill (point \(M\)) to the point on the ground (point \(B\)) is \(250\) meters.

### Final Answer:
The distance of the point from the base of the hill is **250 meters**.