A person observes that the angle of elevation of the top of a pole of height 15 m is `30^(@)`. What is the distance (in metres) of the person from the pole?

Options

A. 15
B. `15sqrt""3`
C. `15//sqrt""3`
D. 30

Answer

To solve the problem, we can follow these steps:

### Step 1: Understand the problem
We need to find the distance of a person from a pole given that the height of the pole is 15 meters and the angle of elevation to the top of the pole is 30 degrees.

### Step 2: Draw a diagram
Visualize the scenario:
- Draw a vertical line representing the pole (height = 15 m).
- From the top of the pole, draw a line at an angle of 30 degrees down to the horizontal ground where the person is standing.
- The horizontal distance from the base of the pole to the person is what we need to find.

### Step 3: Identify the right triangle
In this scenario, we can form a right triangle:
- The height of the pole (15 m) is the opposite side.
- The distance from the person to the pole is the adjacent side.
- The angle of elevation (30 degrees) is between the line of sight to the top of the pole and the horizontal ground.

### Step 4: Use the tangent function
The tangent of an angle in a right triangle is defined as the ratio of the opposite side to the adjacent side. Thus, we can write:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
For our problem:
\[ \tan(30^\circ) = \frac{15}{d} \]
where \(d\) is the distance from the person to the pole.

### Step 5: Substitute the value of \(\tan(30^\circ)\)
We know that:
\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \]
Substituting this into the equation gives:
\[ \frac{1}{\sqrt{3}} = \frac{15}{d} \]

### Step 6: Cross-multiply to solve for \(d\)
Cross-multiplying gives:
\[ d = 15 \cdot \sqrt{3} \]

### Step 7: Calculate the value of \(d\)
Thus, the distance of the person from the pole is:
\[ d = 15\sqrt{3} \text{ meters} \]

### Final Answer
The distance of the person from the pole is \(15\sqrt{3}\) meters.

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