Problem 53 - Algebra and Trigonometry

Question

A man is lying on the beach, flying a kite. He holds the end of the kite string at ground level, and estimates the angle of elevation of the kite to be \(50^{\circ} .\) If the string is 450 ft long, how high is the kite above the ground?

Step-by-Step Solution

Answer

The kite is approximately 344.7 feet above the ground.

Step 1: Understanding the Problem

We have a right triangle formed by the man, the kite, and the horizontal plane. The hypotenuse (the string) is 450 ft, and the angle of elevation is 50 degrees. We need to find the height of the kite above the ground, which is the side opposite the angle.

Step 2: Identifying the Trigonometric Function

Based on the triangle, we can use the sine function for this problem. Sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the hypotenuse. So, we will use the formula: \( \sin(50^{\circ}) = \frac{\text{height}}{450} \) to find the height.

Step 3: Solving for the Height

Let's solve the equation \( \sin(50^{\circ}) = \frac{\text{height}}{450} \). First, calculate \( \sin(50^{\circ}) \), which is approximately 0.766. Then, rearrange the equation to find the height: \( \text{height} = 450 \times 0.766 \).

Step 4: Calculating the Final Answer

Multiply 450 by 0.766 to find the height: \( 450 \times 0.766 = 344.7 \). Thus, the kite is approximately 344.7 feet above the ground.

Key Concepts

Angle of Elevation

The angle of elevation is the angle formed between the horizontal plane and the line of sight when you look upward toward an object.
This angle can be quite helpful in real-life situations, especially when you need to measure the height or distance of something from a horizontal baseline.

When standing on flat ground and looking towards something above you, for example a kite in the sky, you naturally tilt your head upwards. This tilt results in the angle of elevation. In our scenario, the man estimated this angle to be 50 degrees.
Understanding this concept is key so you can apply it correctly in various problems:

• It is always measured from the horizontal line. In the case of the kite, the angle is measured from the line parallel to the ground to the line of sight to the kite.
• This angle helps you to use trigonometric functions to solve for unknown side lengths in triangles formed in such contexts.

Recognizing and correctly identifying the angle of elevation in problems can simplify the process of finding solutions to real-world mathematical situations.

Sine Function

The sine function is a fundamental trigonometric function, especially useful in right triangles.
In our exercise, understanding the sine function was crucial to find the height of the kite.

In trigonometry, the sine of an angle is calculated by the ratio of the length of the opposite side to the hypotenuse in a right triangle:

• For angle theta, \( \sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}} \)
• In practical problems like ours, once you know the angle and the hypotenuse, you can calculate the opposite side's length.
• Accurate values of sine can be looked up in trigonometric tables or calculated using a calculator, which simplifies many calculations.

In this case, the sine of 50 degrees was calculated as approximately 0.766. By multiplying this value by the hypotenuse (450 ft), the height of the kite above the ground was found to be approximately 344.7 ft. Using the sine function helps translate angle measurements into meaningful distances.

Right Triangle

A right triangle is a type of triangle that has one angle equal to 90 degrees, known as the right angle.
It is the cornerstone of many trigonometry problems like the one about the kite.

This triangle format allows for various mathematical calculations using trigonometric functions. Here’s a quick breakdown of key aspects about right triangles:

• They always have one 90-degree angle.
• The side opposite the right angle is the hypotenuse, typically the longest side of the triangle.
• The other two sides are known as the adjacent and opposite sides, relative to a given angle.

For our exercise, the right triangle is formed by the kite's string as the hypotenuse. The height difference from the man on the beach to the kite makes up the opposite side, and the ground represents the adjacent side. Recognizing and understanding these parts help in applying trigonometric formulas effectively to find unknown lengths, like the kite's height above the ground.