Problem 22 - Contemporary Precalculus

Question

Alice is flying a kite. Her hand is three feet above ground level and is holding the end of a 300 -foot-long kite string, which makes an angle of \(57^{\circ}\) with the horizontal. How high is the kite above the ground?

Step-by-Step Solution

Answer

Answer: The kite is approximately 254.60 feet above the ground.

Step 1: Draw a diagram of the given problem

Draw a diagram composed of a right triangle, representing the situation described in the problem. Label one angle with \(57^{\circ}\) and the length of the hypotenuse (the string) as 300 feet. Label the height from the ground to the hand that is holding the string as 3 feet.

Step 2: Find the height of the kite with the triangle

In the triangle, sine is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. So, in this case, our sine function is as follows: \(\sin(57^{\circ}) = \frac{\text{kite height}}{\text{length of string}} = \frac{\text{kite height}}{300}.\)

Step 3: Solve for the kite height

To find the kite height, multiply both sides of the sine function by the length of the string (300 feet): \(\text{kite height} = 300 \times \sin(57^{\circ})\) Calculate the sine of \(57^{\circ}\) and multiply the result by 300: \(\text{kite height} \approx 300 \times 0.83867 \approx 251.60\) feet.

Step 4: Add the height of Alice's hand from the ground

Lastly, we need to add the height of Alice's hand from the ground to find the total height of the kite above the ground: \(\text{total kite height} = \text{kite height} + \text{hand height} = 251.60 + 3 = 254.60\) feet. So, the kite is approximately 254.60 feet above the ground.

Key Concepts

Right Triangle Trigonometry

Right triangle trigonometry is a fundamental aspect of geometry involving the relationships between the angles and sides of right triangles. In these triangles, one of the angles is always equal to 90 degrees, which is the right angle. The two sides that form the right angle are known as the 'legs,' while the side opposite to the right angle is called the 'hypotenuse' - the longest side of the triangle.

Several ratios known as trigonometric functions are used to relate the angles to the lengths of the sides. The primary functions are sine (sin), cosine (cos), and tangent (tan), each representing a specific ratio of side lengths. These functions are essential when solving problems that involve right-angled triangles, like the one where Alice is flying a kite and we need to determine the height of the kite above the ground using the length of the kite string and the angle of elevation.

Sine Function

The sine function is one of the basic trigonometric functions and is represented as sin(θ), where θ is an angle in a right triangle. The sine of an angle is the ratio of the length of the side opposite to the angle (opposite side) over the length of the hypotenuse. The formula is given by:
\[\sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}}\].

In Alice's kite problem, after calculating the sine of the given angle of elevation, we can determine how high the kite is above the ground by using the length of the kite string. This illustrates how the sine function bridges the gap between an angle measure and a side length in trigonometry, providing a practical tool for solving real-world problems where direct measurement is not possible.

Angle of Elevation

The angle of elevation is the angle formed by the line of sight of an observer looking at an object above the horizontal level. It is the angle between the horizontal and the line from the observer's eye to the object. In the context of trigonometry problems, it often plays a crucial role when solving for unknown heights or distances.

In the kite example, the angle of elevation is the angle at which Alice's eyes would follow the string up to the kite in the sky, from the horizontal. This angle helps us use trigonometric functions like the sine to calculate how high the kite is above Alice's head. Remember, it's essential to consider the height at which the string is held above the ground level to get the accurate total height of the kite in real-world situations, as we did by adding the 3 feet from Alice's hand to the ground.