A kite is flying in the sky. The length of the string between a point on the level ground and the kite is 420 m. The angle of elevation that the string makes with the ground is 30°. Assuming that the string is straight and taut with no slack, what is the height of the kite above the ground (in metres)?

Introduction / Context:
This problem involves a simple right angled triangle formed by the kite, the point on the ground directly below the kite, and the point where the person is holding the string. The length of the string and the angle it makes with the ground are given. Assuming the string is straight and taut, the string itself becomes the hypotenuse of the right triangle, while the height of the kite is the side opposite the given angle of elevation. Using basic trigonometry, we can directly compute the vertical height of the kite above the ground.

Given Data / Assumptions:

- Length of the kite string (hypotenuse) is 420 m. - Angle of elevation of the string with the horizontal ground is 30°. - The string is taut and straight, so there is no slack or sag. - The ground between the observer and the point directly below the kite is level.

Concept / Approach:
We consider a right angled triangle where the hypotenuse is the string of length 420 m, the side adjacent to the 30° angle is the horizontal distance on the ground, and the side opposite the 30° angle is the height of the kite. The appropriate trigonometric ratio is sine, since sin of an angle is equal to the length of the opposite side divided by the length of the hypotenuse. Thus, if the height is h, then sin 30° = h / 420. Since sin 30° has a simple standard value, this calculation is straightforward.

Step-by-Step Solution:
Step 1: Let h be the height of the kite above the ground. Step 2: The string of length 420 m is the hypotenuse of the right angled triangle. Step 3: Using the sine ratio, sin 30° = opposite / hypotenuse = h / 420. Step 4: The standard value is sin 30° = 1 / 2. Step 5: Substitute sin 30° into the equation: 1 / 2 = h / 420. Step 6: Multiply both sides by 420 to get h = 420 * (1 / 2) = 210. Step 7: Therefore, the height of the kite is 210 m.

Verification / Alternative check:
To verify, we can also check using the cosine ratio to find the horizontal distance and see if it forms a consistent triangle. If h = 210 m and the hypotenuse is 420 m, then the horizontal distance d satisfies d^2 + h^2 = 420^2. Using cos 30° = √3 / 2, we get d = 420 * (√3 / 2) = 210√3. Now check: d^2 + h^2 = (210√3)^2 + 210^2 = 210^2 * 3 + 210^2 = 210^2 * 4 = (210 * 2)^2 = 420^2, which confirms that the triangle is consistent and that h = 210 m is correct.

Why Other Options Are Wrong:
Option B (140√3) would be larger than 210 and does not match sin 30° = 1 / 2 when used with a 420 m hypotenuse. Option C (210√3) is even larger and corresponds to the horizontal distance, not the vertical height. Option D (150) is less than half of 420 and does not satisfy sin 30° = 1 / 2. Only 210 m makes the sine ratio exact and consistent with the given angle and hypotenuse length.

Common Pitfalls:
Students sometimes confuse sine and cosine and mistakenly compute h using cos 30° instead of sin 30°, which would actually give the horizontal distance rather than the height. Another common error is treating 420 m as the horizontal distance instead of the hypotenuse. Finally, using approximate decimal values for sine and cosine without understanding which side is opposite and which is adjacent can lead to incorrect results. Remembering that sin θ = opposite / hypotenuse and cos θ = adjacent / hypotenuse and clearly labelling the triangle helps avoid these mistakes.

Final Answer:
The height of the kite above the ground is 210 m, which corresponds to option A.