The string of a kite is 250 m long and it makes an angle of `60^(@)` with the horizontal. Find the height of the kite is assuming that there is no slackness in the string.

Answer

To find the height of the kite, we can use the properties of right triangles and trigonometric ratios. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We have a kite that is 250 meters above the ground, and the string makes an angle of 60 degrees with the horizontal. We need to find the height of the kite above the ground.

### Step 2: Identify the Right Triangle
We can visualize the situation as a right triangle:
- Let point A be the point where the kite is located.
- Let point B be the point on the ground directly below the kite.
- Let point C be the point where the string touches the ground.

In this triangle:
- AC is the length of the string (hypotenuse) = 250 m
- AB is the height of the kite (perpendicular) = h (which we need to find)
- Angle CAB = 60 degrees

### Step 3: Use the Sine Function
In a right triangle, the sine of an angle is defined as the ratio of the length of the opposite side (perpendicular) to the length of the hypotenuse. Thus, we can write:

\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]

For our triangle:

\[ \sin(60^\circ) = \frac{h}{250} \]

### Step 4: Substitute the Value of Sine
We know that:

\[ \sin(60^\circ) = \frac{\sqrt{3}}{2} \]

So we can substitute this value into our equation:

\[ \frac{\sqrt{3}}{2} = \frac{h}{250} \]

### Step 5: Cross-Multiply to Solve for h
Cross-multiplying gives us:

\[ h = 250 \cdot \frac{\sqrt{3}}{2} \]

### Step 6: Simplify the Expression
Now, simplify the expression:

\[ h = 125\sqrt{3} \]

### Step 7: Conclusion
Thus, the height of the kite above the ground is:

\[ h \approx 125 \cdot 1.732 \approx 216.5 \text{ meters} \]

### Final Answer
The height of the kite is approximately **216.5 meters**.

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