Problem 32 - Contemporary Precalculus

Question

A rocket shoots straight up from the launchpad. Five seconds after liftoff, an observer two miles away notes that the rocket's angle of elevation is \(3.5^{\circ} .\) Four seconds later, the angle of elevation is \(41^{\circ} .\) How far did the rocket rise during those four seconds?

Step-by-Step Solution

Answer

Answer: The rocket rose \(\Delta h_{ft}\) feet during those four seconds.

Step 1: Initialize given details

We are given: observer-rocket distance: 2 miles initial angle of elevation: \(3.5^{\circ}\) final angle of elevation: \(41^{\circ}\)

Step 2: Convert angles to radians

To make calculations easier, we need to convert angles from degrees to radians. Initial angle in radians: \(3.5^{\circ} * \frac{\pi}{180} = \frac{7}{360}*\pi\) radians Final angle in radians: \(41^{\circ} * \frac{\pi}{180} = \frac{41}{180}*\pi\) radians

Step 3: Find the initial and final heights

Using the tangent function, we can calculate the initial and final heights (in miles) of the rocket h1 and h2. Initial height (h1): \(h1 = 2 * \tan(\frac{7}{360}*\pi)\) Final height (h2): \(h2 = 2 * \tan(\frac{41}{180}*\pi)\)

Step 4: Calculate the height difference

Now, we find the difference between h1 and h2, which will give us the height change during the four seconds interval. Height difference: \(\Delta h = h2 - h1 = 2(\tan(\frac{41}{180}*\pi) - \tan(\frac{7}{360}*\pi))\)

Step 5: Convert the height difference to feet

Since the distance is given in miles, we need to convert the height difference to feet to get the final answer. Height difference in feet: \(\Delta h_{ft} = \Delta h * 5280\) The rocket rose \(\Delta h_{ft}\) feet during those four seconds.

Key Concepts

Angle of Elevation

The angle of elevation is a key concept in trigonometry, particularly useful when determining heights or distances of objects. It is the angle formed between the horizontal line of sight and the line of sight up to an object.
In the context of the exercise, the observer notes angles of elevation at different times to track the rocket's ascent.

• Initial angle of elevation at 5 seconds: \(3.5^{\circ} \)
• Final angle of elevation at 9 seconds: \(41^{\circ} \)

These angles provide crucial information needed to calculate the height the rocket has reached over time. By using these angles, paired with the distance to the observer at ground level, more precise calculations can be made using trigonometric functions.

Tangent Function

The tangent function is a fundamental component in the solution of this rocket problem. It is one of the basic trigonometric functions, often abbreviated as \(\tan\).
The tangent function relates the angle in a right triangle to the ratio of the opposite side to the adjacent side.
When observing the rocket's angle of elevation, the tangent function helps find the height by considering

• the angle of elevation
• the horizontal distance from the observer to the rocket's vertical line of ascent.

Mathematically, this is expressed as:\[ h = d \times \tan(\theta) \] where \( h \) is the height, \( d \) is the distance from the observer, and \( \theta \) is the angle of elevation. This straightforward relationship makes it particularly convenient when working with angles of elevation.

Radians to Degrees Conversion

Radians and degrees are two units of measuring angles. Often, trigonometric calculations are simpler when angles are expressed in radians.
In problems that involve trigonometric functions, you might need to convert

• degrees to radians: multiply by \(\frac{\pi}{180}\)
• radians to degrees: multiply by \(\frac{180}{\pi}\)

In this exercise, the angles given in degrees were converted into radians to facilitate the use of trigonometric functions in calculations, resulting in:

• Initial angle \(3.5^{\circ} = \frac{7}{360} \times \pi \text{ radians} \)
• Final angle \(41^{\circ} = \frac{41}{180} \times \pi \text{ radians} \)

This conversion step is crucial as it aligns with most trigonometric tables and calculators which use radians as their default unit.

Height Calculation

Calculating height in problems involving angles of elevation often involves several steps with trigonometric functions. In our exercise, the height difference during the 4-second interval is found using the tangent function and trigonometric angle conversion.
First, calculate the initial and final height of the rocket using the tangent function:

• Initial height \( h_1 = 2 \times \tan\left( \frac{7}{360} \times \pi \right) \)
• Final height \( h_2 = 2 \times \tan\left( \frac{41}{180} \times \pi \right) \)

Next, find the difference \( \Delta h \) in heights:\[ \Delta h = h_2 - h_1 \]This gives the vertical rise in miles. To convert this to feet for a more conventional measurement, multiply by the conversion factor (1 mile = 5280 feet). Thus, the height difference in feet is:\[ \Delta h_{ft} = \Delta h \times 5280 \]This calculation provides the total height the rocket rose during that four-second period, giving a practical application of trigonometry in determining real-world movements.