A pole casts a shadow of length `2sqrt(3)` m on the ground when the sun's elevation is `60^(@)`. The height of the pole is

Options

A. `4sqrt(3) m`
B. `6 m`
C. `12 m`
D. `3 m`

Explanation

To find the height of the pole, we can use the relationship between the height of the pole, the length of the shadow, and the angle of elevation of the sun. Here's a step-by-step solution:

### Step 1: Understand the problem
We have a pole that casts a shadow of length \(2\sqrt{3}\) meters when the sun's elevation is \(60^\circ\). We need to find the height of the pole.

### Step 2: Set up the right triangle
In this scenario, we can visualize a right triangle where:
- The height of the pole is the opposite side (let's denote it as \(h\)).
- The length of the shadow is the adjacent side, which is \(2\sqrt{3}\) meters.
- The angle of elevation is \(60^\circ\).

### Step 3: Use the tangent function
The tangent of the angle of elevation can be expressed as:
\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \]
For our case:
\[ \tan(60^\circ) = \frac{h}{2\sqrt{3}} \]

### Step 4: Find the value of \(\tan(60^\circ)\)
From trigonometric values, we know:
\[ \tan(60^\circ) = \sqrt{3} \]

### Step 5: Substitute and solve for \(h\)
Now we can substitute \(\tan(60^\circ)\) into our equation:
\[ \sqrt{3} = \frac{h}{2\sqrt{3}} \]

### Step 6: Cross-multiply to solve for \(h\)
Cross-multiplying gives us:
\[ h = 2\sqrt{3} \cdot \sqrt{3} \]

### Step 7: Simplify the expression
Now, simplify the expression:
\[ h = 2 \cdot 3 = 6 \]

### Conclusion
Thus, the height of the pole is:
\[ \boxed{6 \text{ meters}} \]

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