A kite is flying at a height of 123 m. The thread attached to it is assumed to bestretched straight and makes an angle of 60 degree with the level ground. The length ofthe string is (nearest to a whole number):

Options

A. 140 m
B. 139 m
C. 142 m
D. 138 m

Answer

To solve the problem, we need to find the length of the string (hypotenuse) attached to the kite flying at a height of 123 m, making an angle of 60 degrees with the ground. We can use trigonometric ratios to find this length.

### Step-by-Step Solution:

1. **Identify the Triangle**:
We have a right triangle where:
- The height of the kite (opposite side) = 123 m
- The angle with the ground = 60 degrees
- The length of the string (hypotenuse) = x

2. **Use the Sine Function**:
We can use the sine function, which relates the opposite side and the hypotenuse in a right triangle:
\[ \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \]
Here, \(\theta = 60^\circ\), opposite = 123 m, and hypotenuse = x.
\[ \sin(60^\circ) = \frac{123}{x} \]

3. **Substitute the Value of Sine**:
The value of \(\sin(60^\circ)\) is \(\frac{\sqrt{3}}{2}\). Thus, we can write:
\[ \frac{\sqrt{3}}{2} = \frac{123}{x} \]

4. **Cross-Multiply to Solve for x**:
Cross-multiplying gives us:
\[ \sqrt{3} \cdot x = 2 \cdot 123 \]
\[ \sqrt{3} \cdot x = 246 \]

5. **Isolate x**:
Now, divide both sides by \(\sqrt{3}\):
\[ x = \frac{246}{\sqrt{3}} \]

6. **Rationalize the Denominator**:
To simplify, we can multiply the numerator and denominator by \(\sqrt{3}\):
\[ x = \frac{246 \cdot \sqrt{3}}{3} = 82 \cdot \sqrt{3} \]

7. **Calculate the Approximate Value**:
Using \(\sqrt{3} \approx 1.732\):
\[ x \approx 82 \cdot 1.732 \approx 142.024 \]

8. **Round to the Nearest Whole Number**:
The length of the string, rounded to the nearest whole number, is:
\[ x \approx 142 \]

### Final Answer:
The length of the string is approximately **142 meters**.