A pole cast a shadow of length 20 m on thhe ground , when the sun's elevation is `60^(@)` .Find the height of pole .

Options

A. 30 m
B. 40 m
C. 50 m
D. `20sqrt(3)m`

Answer

To find the height of the pole that casts a shadow of length 20 m when the sun's elevation is 60 degrees, we can use trigonometric ratios. Here’s the step-by-step solution:

### Step 1: Understand the Problem
We have a pole (let's call it AB) that casts a shadow (let's call it BC) on the ground. The angle of elevation from the tip of the shadow to the top of the pole is 60 degrees. We need to find the height of the pole (AB).

### Step 2: Draw a Diagram
Draw a right triangle where:
- AB is the height of the pole (which we need to find).
- BC is the length of the shadow (20 m).
- Angle ACB is the angle of elevation (60 degrees).

### Step 3: Identify the Trigonometric Ratio
In right triangle ABC:
- The height of the pole (AB) is the opposite side to angle ACB.
- The length of the shadow (BC) is the adjacent side to angle ACB.

We can use the tangent function, which is defined as:
\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \]
For our triangle:
\[ \tan(60^\circ) = \frac{AB}{BC} \]

### Step 4: Substitute Known Values
We know:
- \( BC = 20 \, \text{m} \)
- \( \tan(60^\circ) = \sqrt{3} \)

Substituting these values into the equation gives:
\[ \sqrt{3} = \frac{AB}{20} \]

### Step 5: Solve for the Height of the Pole (AB)
To find AB, we can rearrange the equation:
\[ AB = 20 \cdot \sqrt{3} \]

### Step 6: Calculate the Height
Now, we can calculate the height:
\[ AB \approx 20 \cdot 1.732 \approx 34.64 \, \text{m} \]

### Final Answer
The height of the pole is approximately **34.64 meters**.

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