A kite is flying at a height of `50sqrt(3)` m from the horizontal. It is attached with a string and makes an angle `60^(@)` with the horizontal. Find the length of the string.

Answer

To find the length of the string attached to the kite, we can use the properties of a right triangle formed by the height of the kite, the horizontal distance, and the string itself. Here’s the step-by-step solution:

### Step 1: Understand the Problem
We know that the kite is flying at a height of \(50\sqrt{3}\) meters from the horizontal and makes an angle of \(60^\circ\) with the horizontal. We need to find the length of the string, which acts as the hypotenuse of the right triangle formed.

### Step 2: Draw the Right Triangle
1. Draw a horizontal line to represent the ground.
2. From a point on this line, draw a vertical line up to a height of \(50\sqrt{3}\) meters. This represents the height of the kite.
3. Connect the top of this vertical line to the point on the horizontal line where the string is attached, forming a right triangle.

### Step 3: Label the Triangle
- Let the point where the kite is located be point A (top of the vertical line).
- Let the point on the ground directly below the kite be point B.
- Let the point where the string is attached to the ground be point C.
- The height \(AB = 50\sqrt{3}\) m.
- The angle \(CAB = 60^\circ\).
- The length of the string \(AC\) is what we need to find.

### Step 4: Use the Sine Function
In the right triangle \(ABC\):
- The sine of angle \(C\) (which is \(60^\circ\)) is given by the formula:
\[ \sin(C) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{AB}{AC} \]
Here, \(AB\) is the height \(50\sqrt{3}\) m and \(AC\) is the length of the string \(x\).

### Step 5: Set Up the Equation
Substituting the known values into the sine formula:
\[ \sin(60^\circ) = \frac{50\sqrt{3}}{x} \]
We know that \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\).

### Step 6: Solve for \(x\)
Substituting \(\sin(60^\circ)\) into the equation:
\[ \frac{\sqrt{3}}{2} = \frac{50\sqrt{3}}{x} \]
Cross-multiplying gives:
\[ \sqrt{3} \cdot x = 100\sqrt{3} \]
Dividing both sides by \(\sqrt{3}\):
\[ x = 100 \]

### Final Answer
The length of the string is \(100\) meters.

---