A kite is flying in the sky. The length of string between a point on the ground and kite is 420 m. The angle of elevation of string with the ground is `30^(@)` Assuming that there is no slack in the string, then what is the height (in metres) of the kite?

Options

A. 210
B. `140 sqrt""3`
C. `10 sqrt""3`
D. 150

Answer

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

### Step 1: Understand the problem
We have a kite flying at a height, and the string from the ground to the kite forms an angle of elevation of \(30^\circ\) with the ground. The length of the string (hypotenuse) is 420 m.

### Step 2: Identify the right triangle
We can visualize the situation as a right triangle where:
- The height of the kite from the ground is the opposite side (let's call it \(h\)).
- The length of the string is the hypotenuse (420 m).
- The angle of elevation is \(30^\circ\).

### 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:
\[ \sin(30^\circ) = \frac{h}{420} \]

### Step 4: Substitute the known value
We know that \(\sin(30^\circ) = \frac{1}{2}\). Substituting this value into the equation gives:
\[ \frac{1}{2} = \frac{h}{420} \]

### Step 5: Solve for \(h\)
To find \(h\), we can rearrange the equation:
\[ h = 420 \times \frac{1}{2} \]
\[ h = \frac{420}{2} = 210 \text{ m} \]

### Conclusion
The height of the kite is \(210\) meters.

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