At 129 metre away from the foot of a cliff on level of ground , the angle of elevation of the top of the cliff is `30^(@)` . The height of this cliff is :

Options

A. `50sqrt(3)` metre
B. `45sqrt(3)` metre
C. `43sqrt(3)` metre
D. `47sqrt(3)` metre

Answer

To solve the problem, we will use trigonometric ratios. Here's a step-by-step solution:

### Step 1: Understand the Problem
We are given a cliff and the distance from the foot of the cliff to the point of observation is 129 meters. The angle of elevation to the top of the cliff is 30 degrees. We need to find the height of the cliff.

### Step 2: Draw a Diagram
Draw a right triangle where:
- The horizontal leg (base) represents the distance from the foot of the cliff, which is 129 meters.
- The vertical leg (perpendicular) represents the height of the cliff (let's denote it as \( h \)).
- The angle of elevation from the ground to the top of the cliff is 30 degrees.

### Step 3: Use the Tangent Function
In a right triangle, the tangent of an angle is defined as the ratio of the opposite side (height of the cliff) to the adjacent side (distance from the foot of the cliff). Therefore, we can write:

\[ \tan(30^\circ) = \frac{h}{129} \]

### Step 4: Substitute the Value of \( \tan(30^\circ) \)
We know that:

\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \]

Substituting this value into the equation gives us:

\[ \frac{1}{\sqrt{3}} = \frac{h}{129} \]

### Step 5: Solve for \( h \)
To find \( h \), we can cross-multiply:

\[ h = 129 \cdot \frac{1}{\sqrt{3}} \]

This simplifies to:

\[ h = \frac{129}{\sqrt{3}} \]

### Step 6: Rationalize the Denominator
To simplify \( h \), we can multiply the numerator and denominator by \( \sqrt{3} \):

\[ h = \frac{129 \cdot \sqrt{3}}{3} \]

Calculating this gives:

\[ h = 43 \sqrt{3} \]

### Step 7: Final Answer
The height of the cliff is \( 43 \sqrt{3} \) meters.

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