Problem 25 - Precalculus: Functions and Graphs

Question

A person flying a kite holds the string 4 feet above ground level. The string of the kite is taut and makes an angle of \(60^{\circ}\) with the horizontal (see the figure). Approximate the height of the kite above level ground if 500 feet of string is payed out. (IMAGE CAN NOT COPY)

Step-by-Step Solution

Answer

The height of the kite above ground is approximately 437 feet.

Step 1: Identify the Triangle

In this scenario, a right triangle is formed with the string as the hypotenuse. The altitude of the kite forms the opposite side of the angle, and the ground projection forms the adjacent side.

Step 2: Write the Sine Function Formula

To find the height of the kite above the ground, we use the sine function. The sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse. Therefore, we write:\[\sin(60^\circ) = \frac{\text{Height of the kite above string level}}{500}\]

Step 3: Calculate the Height Above String Level

Solve for the height above string level using the sine value:\[\sin(60^\circ) = \frac{\sqrt{3}}{2} \\frac{\sqrt{3}}{2} = \frac{h}{500} \\Rightarrow h = 500 \times \frac{\sqrt{3}}{2} \\Rightarrow h \approx 500 \times 0.866 = 433 \text{ feet}\]

Step 4: Add the String Offset Height

Since the string is held 4 feet above the ground, the total height of the kite above ground level is:\[433 + 4 = 437 \text{ feet}\]

Key Concepts

Right Triangle

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. This right angle creates two sides that meet perpendicular to each other. In the context of our kite flying exercise, the string forms the hypotenuse, which is the side opposite the right angle.
This configuration helps in visualizing the relationships between the sides, especially when using trigonometric functions like sine.
To identify a right triangle in a problem, look for the following traits:

• One angle is 90 degrees.
• The longest side, known as the hypotenuse, is always opposite the right angle.
• The other two sides are called legs; one being adjacent and the other opposite to the angle of interest, often utilized in trigonometry.

The right triangle in this problem helps us apply trigonometric functions to find unknown side lengths, like the height of the kite above the ground.

Sine Function

The sine function is fundamental in trigonometry and deals with the relationship between angles and sides in a right triangle. Specifically, sine of an angle is the ratio of the length of the opposite side to the hypotenuse. For example, for an angle \( \theta \) in a right triangle, the sine function is given by:
\[ \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \].
This makes it useful in many real-world problems, including calculating heights and distances.
In our exercise, the angle of elevation is \(60^{\circ}\), the hypotenuse (string length) is 500 feet, and the opposite side is the height of the kite above string level. By applying the sine function, we calculate: \( \sin(60^{\circ}) = \frac{\text{Height}}{500} \). Understanding the sine function can help you easily solve similar problems where such a relationship can illustrate how elements of the triangle relate to one another.

Angle of Elevation

The angle of elevation is an important concept in trigonometry that refers to the angle formed between the horizontal line and the line of sight when looking at an object above the horizontal. In the context of our exercise, it is the angle formed by the kite string and the ground.
Identifying this angle involves the following steps:

• The horizontal line acts as the base of the right triangle - in this case, the ground level.
• The line of sight or the string forms the hypotenuse.

The angle of elevation, in our case \(60^{\circ}\), helps you apply trigonometric functions effectively. By determining this angle, you can calculate heights and distances that might initially seem challenging. Remember, understanding angles of elevation can greatly assist in visual scenarios involving heights or inclined paths.