From a point of observation at the top of a 175 m high cliff, the angles of depression of two objects are `x ^(@) and y ^(@)` such that ` tan x ^(@) = 2.5 and tan y ^(@) =1.4.` If the point of observation and the two objects are long the same straight line, find the distance betwen the two objects if they are on the :
opposite sides of the cliff.

Answer

To solve the problem, we need to find the distance between two objects located on opposite sides of a cliff, given the height of the cliff and the angles of depression to the objects.

### Step-by-Step Solution:

1. **Identify the given values**:
- Height of the cliff (AB) = 175 m
- \( \tan x = 2.5 \)
- \( \tan y = 1.4 \)

2. **Set up the triangles**:
- Let point A be the top of the cliff, point B be the base of the cliff, point C be the position of the first object, and point D be the position of the second object.
- The angles of depression from A to C and A to D are \( x \) and \( y \) respectively.

3. **Using the tangent function for triangle ABC**:
- In triangle ABC, \( \tan x = \frac{AB}{BC} \)
- Substituting the known values:
\[ 2.5 = \frac{175}{BC} \]
- Rearranging to find BC:
\[ BC = \frac{175}{2.5} \]
- Calculating BC:
\[ BC = 70 \text{ m} \]

4. **Using the tangent function for triangle ABD**:
- In triangle ABD, \( \tan y = \frac{AB}{BD} \)
- Substituting the known values:
\[ 1.4 = \frac{175}{BD} \]
- Rearranging to find BD:
\[ BD = \frac{175}{1.4} \]
- Calculating BD:
\[ BD = 125 \text{ m} \]

5. **Finding the total distance between the two objects (CD)**:
- Since the objects are on opposite sides of the cliff, the total distance CD is:
\[ CD = BC + BD \]
- Substituting the values:
\[ CD = 70 + 125 = 195 \text{ m} \]

### Final Answer:
The distance between the two objects is **195 meters**.