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 :
same side of the cliff

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Understand the Problem
We have a cliff of height 175 m. From the top of the cliff, we are given two angles of depression, \( x^\circ \) and \( y^\circ \), with their respective tangents:
- \( \tan x^\circ = 2.5 \)
- \( \tan y^\circ = 1.4 \)

We need to find the distance between two objects, C and D, located at the angles of depression \( x^\circ \) and \( y^\circ \) respectively.

### Step 2: Identify the Right Triangles
Let:
- Point A be the top of the cliff.
- Point B be the base of the cliff directly below A.
- Point C be the position of the first object at angle \( x^\circ \).
- Point D be the position of the second object at angle \( y^\circ \).

### Step 3: Set Up the Right Triangles
In triangle ABC:
- \( AB = 175 \) m (height of the cliff)
- \( BC \) is the horizontal distance from the base of the cliff to object C.

Using the tangent function:
\[ \tan x^\circ = \frac{AB}{BC} \implies BC = \frac{AB}{\tan x^\circ} = \frac{175}{2.5} \]

### Step 4: Calculate \( BC \)
Calculating \( BC \):
\[ BC = \frac{175}{2.5} = \frac{1750}{25} = 70 \text{ m} \]

### Step 5: Set Up the Second Right Triangle
In triangle ADB:
- \( AD = 175 \) m (same height)
- \( BD \) is the horizontal distance from the base of the cliff to object D.

Using the tangent function:
\[ \tan y^\circ = \frac{AB}{BD} \implies BD = \frac{AB}{\tan y^\circ} = \frac{175}{1.4} \]

### Step 6: Calculate \( BD \)
Calculating \( BD \):
\[ BD = \frac{175}{1.4} = \frac{1750}{14} = 125 \text{ m} \]

### Step 7: Find the Distance Between Objects C and D
The distance \( DC \) between the two objects can be found by subtracting \( BC \) from \( BD \):
\[ DC = BD - BC = 125 - 70 = 55 \text{ m} \]

### Final Answer
The distance between the two objects is **55 meters**.

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