A kite is flying in the sky, attached to a straight string that is pulled taut. The length of the string from a point on the ground to the kite is 420 m, and the angle of elevation of the string with the horizontal ground is 30°. Assuming there is no slack in the string, what is the vertical height (in metres) of the kite above the ground?

Introduction:
This problem is a direct application of basic trigonometry to a height and distance context involving a flying kite. The string forms the hypotenuse of a right-angled triangle, and the angle between the string and the horizontal ground is given. We must find the vertical height of the kite using the sine function.

Given Data / Assumptions:

• Length of the string (hypotenuse) = 420 m.
• Angle of elevation between string and ground = 30°.
• The string is taut (no slack), so it forms a straight line.
• The ground is level and the kite is directly above some point on the ground.

Concept / Approach:
In a right-angled triangle, the sine of an angle is defined as opposite side divided by hypotenuse. Here, the opposite side is the vertical height h of the kite, and the hypotenuse is the string length 420 m. Thus:
sin 30° = h / 420.We use the known value sin 30° = 1/2 to solve for h.

Step-by-Step Solution:
Step 1: Write the sine relation: sin 30° = h / 420.Step 2: Substitute sin 30° = 1/2, giving 1/2 = h / 420.Step 3: Solve for h: h = 420 * (1/2) = 210 m.

Verification / Alternative check:
We can also find the horizontal distance to the point directly under the kite: using cos 30° = adjacent / hypotenuse, we get adjacent = 420 * cos 30° = 420 * (√3/2) = 210√3 m. This is consistent with a standard 30°–60°–90° triangle, where the sides are in the ratio 1 : √3 : 2. Thus, our value h = 210 m fits the triangle perfectly.

Why Other Options Are Wrong:
140√3 and 210√3 represent different legs of a 30°–60°–90° triangle but not the vertical height for a 30° angle with hypotenuse 420. Heights like 150 m or 120 m do not satisfy sin 30° = h/420 when checked: 150/420 and 120/420 are not equal to 1/2.

Common Pitfalls:
A common mistake is to mistakenly use cosine instead of sine for the vertical height, or to confuse which side (opposite or adjacent) corresponds to the given angle. Remember: with the angle at the ground and the vertical height opposite that angle, sine is the correct ratio to use.

Final Answer:
The height of the kite above the ground is 210 m.