A kite is attached to a 100 m long string. Find the greatest height reached by the kite when its string makes an angle of `60^(@)` with the travel round.

Answer

To find the greatest height reached by the kite when its string makes an angle of \(60^\circ\) with the ground, we can use trigonometric principles. Here’s a step-by-step solution:

### Step 1: Understand the problem
We have a kite attached to a 100 m long string, and the string makes an angle of \(60^\circ\) with the horizontal ground. We need to find the vertical height (h) of the kite above the ground.

### Step 2: Visualize the scenario
Draw a right triangle where:
- The hypotenuse (the string) is 100 m.
- The angle with the horizontal (ground) is \(60^\circ\).
- The height (h) is the opposite side to the angle \(60^\circ\).

### 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 to the length of the hypotenuse. Therefore, we can write:

\[ \sin(60^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{h}{100} \]

### Step 4: Substitute the known values
We know that \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\). Substituting this into the equation gives:

\[ \frac{\sqrt{3}}{2} = \frac{h}{100} \]

### Step 5: Solve for h
To find h, we can rearrange the equation:

\[ h = 100 \cdot \frac{\sqrt{3}}{2} \]

Calculating this gives:

\[ h = 50\sqrt{3} \]

### Step 6: Calculate the numerical value
Using the approximate value of \(\sqrt{3} \approx 1.732\):

\[ h \approx 50 \cdot 1.732 \approx 86.6 \text{ m} \]

### Conclusion
The greatest height reached by the kite is approximately \(86.6\) meters.

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