The angle of elevation from a point on the bank of a river of the top of a temple on the other bank is `45^(@)` Retreating 50m, the observer finds the new angle of elevation as `30^(@).` What is the width of the river?

Options

A. 50 m
B. `50 sqrt3`m
C. `50 // (sqrt3-1) m `
D. `100 m`

Answer

To solve the problem step by step, we will use trigonometric principles involving right triangles formed by the observer's position, the height of the temple, and the width of the river.

### Step 1: Define the Variables
Let:
- \( W \) = width of the river (in meters)
- \( h \) = height of the temple (in meters)

### Step 2: Analyze the First Position
From the first position (point A), the angle of elevation to the top of the temple is \( 45^\circ \).

Using the tangent function:
\[ \tan(45^\circ) = \frac{h}{W} \]
Since \( \tan(45^\circ) = 1 \):
\[ 1 = \frac{h}{W} \implies h = W \quad \text{(Equation 1)} \]

### Step 3: Analyze the Second Position
After retreating 50 meters to point B, the angle of elevation to the top of the temple is \( 30^\circ \).

Using the tangent function again:
\[ \tan(30^\circ) = \frac{h}{50 + W} \]
Since \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \):
\[ \frac{1}{\sqrt{3}} = \frac{h}{50 + W} \]
Cross-multiplying gives:
\[ h = \frac{50 + W}{\sqrt{3}} \quad \text{(Equation 2)} \]

### Step 4: Substitute Equation 1 into Equation 2
From Equation 1, we have \( h = W \). Substitute \( W \) for \( h \) in Equation 2:
\[ W = \frac{50 + W}{\sqrt{3}} \]

### Step 5: Solve for W
Multiply both sides by \( \sqrt{3} \):
\[ W \sqrt{3} = 50 + W \]
Rearranging gives:
\[ W \sqrt{3} - W = 50 \]
Factoring out \( W \):
\[ W (\sqrt{3} - 1) = 50 \]
Now, solve for \( W \):
\[ W = \frac{50}{\sqrt{3} - 1} \]

### Final Answer
The width of the river is:
\[ W = \frac{50}{\sqrt{3} - 1} \text{ meters} \]

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