Problem 53 - Precalculus Mathematics for Calculus

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 \(450 \mathrm{ft}\) long, how high is the kite above the ground?

Step-by-Step Solution

Answer

The kite is approximately 344.7 ft above the ground.

Step 1: Identify the Right Triangle

First, understand the problem by recognizing the right triangle involved. The ground, the vertical height of the kite, and the string form a right triangle.

Step 2: Assign Known Values

Label the known values: the length of the string (hypotenuse) is 450 ft, and the angle of elevation is 50°.

Step 3: Use the Sine Function

The sine of an angle in a right triangle is the opposite side (height of the kite) divided by the hypotenuse (kite string). Use the equation: \( \sin(50^{\circ}) = \frac{\text{height}}{450} \) to solve for the height.

Step 4: Calculate the Height

Rearrange the formula to find the height: \( \text{height} = 450 \times \sin(50^{\circ}) \). Calculate this using a calculator: \( \text{height} \approx 450 \times 0.766 = 344.7 \) ft.

Key Concepts

Angle of Elevation

An angle of elevation is a way to measure how high something is in relation to a horizontal line, like the ground. Imagine standing on the beach looking up at a kite. The angle your eyes make from looking straight ahead to looking up at the kite is the angle of elevation.
This angle is always measured from the horizontal up to the object. In our exercise, the man estimates the angle of elevation to be \(50^{\circ}\).
Some key points about the angle of elevation are:

• It's always measured upwards from a horizontal line.
• It's an angle that is found in real-world applications such as aviation, construction, and various games or sports.
• To estimate these angles accurately in real life, tools like clinometers or smartphones with the right app can be used.

Right Triangle

A right triangle is a triangle in which one angle measures exactly \(90^{\circ}\). This triangle has three sides known as the hypotenuse, opposite, and adjacent.
In the kite flying scenario, the right triangle helps us work out how high the kite is flying. The ground is one side, the string is another, and the height from the ground to the kite is the third side.
Some important aspects of right triangles are:

• The hypotenuse is the longest side, opposite the right angle. For the kite problem, it’s the string length, \(450\) ft.
• The angle opposite the observed side helps determine relationships in trigonometry.
• The right triangle can be used to make calculations about heights and distances that would otherwise be hard to measure directly.

Understanding right triangles is essential for solving problems using trigonometric functions such as sine, cosine, and tangent.

Sine Function

The sine function is one of the basic trigonometric functions used to relate the angles and sides of a right triangle.
For an angle in a right triangle, the sine gives the ratio of the length of the side opposite the angle to the hypotenuse.
In mathematical terms, for an angle \(\theta\), the sine function is given by:\[\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\]In our exercise, this equation helps calculate the height the kite is above the ground. Given the angle of \(50^{\circ}\) and the hypotenuse of \(450\) ft, you use the sine function to find that missing height.
To achieve this:

• Identify the angle and hypotenuse lengths from the problem.
• Use the sine function formula to solve for the opposite side, which is the height.
• Perform the calculations: \(\text{height} = 450 \times \sin(50^{\circ})\).

For angles not typically found on a trigonometric table, we often use a calculator to find sine values, making understanding the sine function essential in finding unknown distances and heights in various practical problems.