Question

Sierra is flying a kite. She has let out 55 feet of string. If the angle of elevation is \(35^{\circ}\) and the hand holding the string is 6 feet from the ground, what is the altitude of the kite? Round to the nearest tenth.

Short answer

The altitude of the kite is approximately 37.5 feet.

Step-by-step solution

Identify the right triangle components

The kite string forms the hypotenuse of a right triangle, where the horizontal leg is along the ground and the vertical leg is the altitude of the kite above the hand holding the string. We are given the length of the hypotenuse (55 feet) and the angle of elevation (35°).

Use the sine function

The sine of an angle in a right triangle is equal to the opposite side (altitude in this case) over the hypotenuse. The formula is \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \). For this problem, \( \theta = 35^{\circ} \) and the hypotenuse is 55 feet.

Substitute and solve for the altitude

Substitute the known values into the sine formula: \( \sin(35^{\circ}) = \frac{h}{55} \), where \( h \) is the altitude of the kite above the hand. Solving for \( h \), we get \( h = 55 \times \sin(35^{\circ}) \). Use a calculator to find the sine value and compute \( h \).

Calculate the numerical value

Using a calculator, compute \( \sin(35^{\circ}) \approx 0.5736 \). Multiply by 55 to find \( h \): \( h \approx 55 \times 0.5736 \approx 31.548 \).

Adjust for the height of the hand

The calculated altitude \( h \) is above the height where the string is held (6 feet above the ground). Therefore, the total altitude of the kite above the ground is \( 31.548 + 6 \approx 37.548 \) feet.

Round the result

Round the total altitude to the nearest tenth, which gives \( 37.5 \) feet.

Key concepts

Right Triangle

In trigonometry, a right triangle is a triangle that has one angle equal to 90 degrees. This forms the basis for many trigonometric calculations since the relationships between the angles and sides are defined by specific functions. When solving problems like finding the altitude of a kite, we use the properties of a right triangle to set up a logical and mathematical framework.

• **Hypotenuse**: This is the longest side opposite the right angle. In this problem, the kite string acts as the hypotenuse.
• **Opposite Side**: This is the side opposite the angle of elevation, which in this case represents the altitude of the kite.
• **Adjacent Side**: This side lies next to the angle of elevation, lying along the ground.

When given two elements, such as an angle and a side, we can determine all other elements using trigonometric ratios, such as the sine function.

Angle of Elevation

The angle of elevation is the angle formed between the line of sight of an observer looking upwards at an object and the horizontal line from the observer. For Sierra flying the kite, the angle of elevation is the angle between the string and the ground. Understanding and identifying this angle is crucial in real-world applications to determine heights or distances.

• It provides a measure of how steeply the observer is looking upwards.
• Angles of elevation are often measured in degrees, which provides a convenient way to work with standard trigonometric functions.
• The angle of elevation is used along with the length of line of sight (hypotenuse) in solving right triangle problems.

The angle helps in forming the required right triangle for problems to allow use of trigonometric functions like sine to find the unknown side lengths.

Sine Function

The sine function is one of the primary trigonometric functions used extensively in relation to circles and right triangles. It is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right triangle. In the kite scenario, sine helps us determine the altitude or the height of the kite above Sierra's hand. Here's how you can utilize the sine function:

• **Formula**: \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \)
• Given: \( \theta = 35^{\circ} \) and hypotenuse = 55 feet, we use set this up as \( \sin(35^{\circ}) = \frac{h}{55} \).
• Solve for the opposite side (kite altitude \( h \)) by rearranging the formula and then solving: \( h = 55 \times \sin(35^{\circ}) \).

Using a calculator to find sine, you multiply by the hypotenuse to find the unknown opposite side. This calculation helps determine how high the kite is flying above your hand's height when it holds the string.