Problem 63 - Algebra and Trigonometry

Question

Height of a Rocket A rocket fired straight up is tracked by an observer on the ground a mile away. (a) Show that when the angle of elevation is \(\theta\) , the height of the rocket in feet is \(h=5280 \tan \theta\) . (b) Complete the table to find the height of the rocket at the given angles of elevation. $$ \begin{array}{|c|c|c|c|c|}\hline \theta & {20^{\circ}} & {60^{\circ}} & {80^{\circ}} & {85^{\circ}} \\ \hline h & {} & {} & {} \\ \hline\end{array} $$

Step-by-Step Solution

Answer

The height formula is \(h=5280 \tan \theta\), and heights are approximately 1920 ft, 9149 ft, 29935 ft, and 60354 ft for the angles given.

Step 1: Understanding the Problem

The problem involves tracking a rocket that is fired straight up. The observer is one mile (5280 feet) from the launch site and measures the angle of elevation \(\theta\) from the ground to the rocket. The goal is to express the height of the rocket \(h\) as a function of \(\theta\), and fill in a table with the rocket's height at specific angles.

Step 2: Expressing Height in Terms of Angle

Since the rocket, the observer, and the line of sight form a right triangle with the distance from the observer to the launch site as the base, the height of the rocket \(h\) is the opposite side relative to the angle \(\theta\). Using trigonometry, the tangent function gives us \(\tan(\theta) = \frac{h}{5280}\). Solving for \(h\), we get \(h = 5280 \tan(\theta)\).

Step 3: Calculating Height for Given Angles

Use the formula \(h = 5280 \tan(\theta)\) to find the height for each given angle. The angles provided are \(20^\circ\), \(60^\circ\), \(80^\circ\), and \(85^\circ\). Use a calculator to evaluate each tangent function.- For \(\theta = 20^\circ\), \(\tan(20^\circ) \approx 0.364\), so \(h = 5280 \times 0.364 \approx 1920\) feet.- For \(\theta = 60^\circ\), \(\tan(60^\circ) = \sqrt{3} \approx 1.732\), so \(h = 5280 \times 1.732 \approx 9148.96\) feet.- For \(\theta = 80^\circ\), \(\tan(80^\circ) \approx 5.671\), so \(h = 5280 \times 5.671 \approx 29934.88\) feet.- For \(\theta = 85^\circ\), \(\tan(85^\circ) \approx 11.430\), so \(h = 5280 \times 11.430 \approx 60354.4\) feet.

Step 4: Completing the Table

Fill in the calculated heights into the table:\[\begin{array}{|c|c|c|c|c|}\hline \theta & {20^{\circ}} & {60^{\circ}} & {80^{\circ}} & {85^{\circ}} \\hline h & {1920} & {9148.96} & {29934.88} & {60354.4} \\hline\end{array}\]

Key Concepts

Tangent Function

The tangent function is a crucial concept in trigonometry used to relate angles in right triangles to the ratios of the sides. It is defined as the ratio of the opposite side to the adjacent side in a right triangle. This function is especially powerful when dealing with angles of elevation or depression.
The tangent function is expressed as \( \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \). This formula helps us find a missing side of a triangle if we know one side and an angle other than the right angle.
In the context of our problem, knowing the angle \( \theta \) and the distance from the observer gives us a direct way to calculate the height of the rocket using the tangent function's principle.

Right Triangle

A right triangle is a triangle where one of the angles is exactly 90 degrees. This type of triangle is the foundation for studying trigonometric ratios, including sine, cosine, and tangent.
In a right triangle:

• The side opposite the right angle is called the hypotenuse.
• The other two sides are known as the legs. The leg opposite the angle of interest is termed the opposite, while the leg adjacent to it is termed the adjacent.

Understanding these components is vital for correctly applying trigonometric functions like tangent.
In our exercise, the observer, the height of the rocket, and the horizontal distance from the observer to the rocket's launch point form a right triangle.

Angle of Elevation

The angle of elevation is the angle formed by the horizontal line and the line of sight to an object above the horizontal. It's measured from the ground up to the line of sight. This angle is crucial when calculating the height of objects using trigonometry.
Consider standing on flat ground observing a rocket. The angle your line of sight forms as you look up at the rocket is the angle of elevation.
This measurement allows you to apply trigonometric calculations, like the tangent function, to find the rocket's height based on the distance to the launch site and the measurable angle of elevation.

Height Calculation

Calculating the height of an object involves using the known distance between the observer and the object and the angle of elevation. The process relies heavily on the tangent function due to the right triangle formed.
When the angle of elevation \( \theta \) is known, you can use the equation \( h = 5280 \tan(\theta) \) to find the rocket's height. In this formula:

• \( h \) represents the height of the rocket,
• 5280 is the distance from the observer to the location of the rocket’s launch (in feet), and
• \( \tan(\theta) \) is the tangent of the angle of elevation.

The calculation steps involve finding the tangent of the given angles, multiplying by the distance, and thus obtaining the height. This systematic approach allows accurate and efficient height determination of the rocket from different angles as outlined in the original solution.