A circus artist is climbing a 20 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground . Find the height of the pole, if the angle and by the rope with ground leave is `30^(@)`

Answer

To find the height of the pole using the information provided, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have a vertical pole and a rope that is tied from the top of the pole to the ground, forming a right triangle. The length of the rope (hypotenuse) is 20 m, and the angle between the rope and the ground is 30 degrees. We need to find the height of the pole (which is the opposite side of the triangle).

2. **Drawing the Diagram**:
Let's label the points:
- Let point A be the top of the pole.
- Let point B be the point on the ground directly below point A.
- Let point C be the point where the rope touches the ground.
- The length of the rope (AC) is 20 m.
- The angle ∠CAB is 30 degrees.
- The height of the pole (AB) is what we need to find, and we will denote it as h.

3. **Using Trigonometric Ratios**:
In triangle ABC, we can use the sine function:
\[ \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \]
Here, θ = 30 degrees, the opposite side is h (height of the pole), and the hypotenuse is AC (length of the rope).

4. **Setting Up the Equation**:
So, we can write:
\[ \sin(30^\circ) = \frac{h}{20} \]
We know that:
\[ \sin(30^\circ) = \frac{1}{2} \]
Therefore, we can substitute this into our equation:
\[ \frac{1}{2} = \frac{h}{20} \]

5. **Cross-Multiplying**:
To solve for h, we can cross-multiply:
\[ 1 \cdot 20 = 2 \cdot h \]
This simplifies to:
\[ 20 = 2h \]

6. **Solving for h**:
Now, divide both sides by 2:
\[ h = \frac{20}{2} = 10 \text{ m} \]

7. **Conclusion**:
The height of the pole is 10 meters.