Question

A kite is flown with a thread of \(250 \mathrm{~m}\) length. If the threats assumed to be stretched and makes an angle of \(60^{\circ}\) withte horizontal, then the height of the kite above the ground (approx.): (a) \(216.25 \mathrm{~m}\) (b) \(215.25 \mathrm{~m}\) (c) \(212.25 \mathrm{~m}\) (d) \(210.25 \mathrm{~m}\)

Short answer

(a) 216.25 m (b) 240 m (c) 200 m (d) 180 m Answer: (a) 216.25 m

Step-by-step solution

Identifying the given values and the unknown height

The length of the kite's thread, which is the hypotenuse of the right triangle, is 250 m. The angle between the thread and the horizontal, which we can denote as θ, is 60°. We need to find the height of the kite above the ground, which can be denoted as h.

Using the sine function

Since we know the length of the hypotenuse and the angle θ, we can use the sine function, which relates the opposite side (height, h), the hypotenuse (the thread's length), and the angle between them. The sine function can be represented as: sin(θ) = opposite side / hypotenuse In our case: sin(60°) = h / 250

Solving for height

Now, we need to solve for h in the equation sin(60°) = h / 250. To do this, multiply both sides by 250: h = 250 × sin(60°)

Calculating the height

We know sin(60°) = √3/2, so the equation becomes: h = 250 × (√3/2) h ≈ 216.506 m

Comparing the result to the answer choices

Comparing the calculated height to the given answer choices, we can see that the value closest to our result is 216.25 m, which corresponds to answer (a). So, the approximate height of the kite above the ground is 216.25 m.

Key concepts

Sine Function Application

The sine function is one of the fundamental tools of trigonometry, particularly when dealing with right triangles. It provides a relationship between angles and side lengths, defining the sine of an angle as the ratio of the length of the side opposite the angle to the length of the hypotenuse.

To apply the sine function, you need to identify which angle you're working with and the sides in relation to that angle. In practice, if you're given a right triangle and you know the length of the hypotenuse (the longest side, opposite the right angle) and an angle, other than the right angle, you can find the length of the side opposite that angle by rearranging the formula:

\( \text{sin}(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}} \).

In the example of the flying kite, knowing the angle of the thread at 60° and the length of the thread (hypotenuse) allows the application of the sine function to find the height of the kite (opposite side). This technique is not only useful for kites but is widely applicable in various fields such as engineering, physics, and even in everyday problem-solving exercises.

Solving Right Triangles

Solving right triangles refers to the process of finding unknown side lengths and angle measures in a right triangle. The sine function is just one part of this process; it is most useful when an acute angle and the hypotenuse are known, and you need to find the opposite side. To solve a right triangle, you can also use the cosine function for adjacent sides or the tangent function for the ratio of the opposite to the adjacent side.

Here are the basic steps for solving a right triangle:

• Identify the known and unknown elements of the triangle.
• Choose the correct trigonometric function based on the given information.
• Rearrange the trigonometric equation and solve for the unknown.
• Calculate the unknown sides or angles using the appropriate trigonometric identities or ratios.

This process can be applied to a diverse array of problems, from simple homework questions to real-world architectural and navigational challenges.

Hypotenuse Calculation

The hypotenuse of a right triangle is the side opposite the right angle and is critical for many trigonometric calculations. Calculating the hypotenuse requires different approaches based on the available information. If you know the lengths of the other two sides, you can use Pythagoras’ theorem. However, if you have one side and an acute angle, trigonometric functions come into play.

For example:
\( \text{if } a \text{ and } b \text{ are the legs, then the hypotenuse } c \text{ is given by } c=\text{sqrt}(a^2 + b^2) \).

On the other hand, using the sine function, as in the kite problem, you start with the angle and the hypotenuse to find an unknown side (\( h = \text{hypotenuse} \times \text{sin}(\theta) \)). Understanding these principles and knowing when to apply them is vital for efficiently solving for the hypotenuse in various mathematical and applied problems.