Problem 67 - College Algebra

Question

Freedom 7 was the spacecraft that carried the first American into space in \(1961 .\) Total flight time was 15 minutes, and the spacecraft reached a maximum height of 116 miles. Consider a function, \(s,\) that expresses Freedom Ts height, \(s(t),\) in miles, after \(t\) minutes. Is \(s\) a one-to-one function? Explain your answer.

Step-by-Step Solution

Answer

Yes, \(s\) is a one-to-one function because at each minute of Freedom 7's flight time, it occupied a unique height.

Step 1: Understand the journey of Freedom 7

Understand that Freedom 7's flight had a ballistic trajectory, which means it ascended, reached a maximum height, then descended. It did not stay at the same height for any duration of time. So, at different times, the spacecraft was at different heights.

Step 2: Apply the definition of a one-to-one function

Know that a function \(s(t)\) is one-to-one if, for any two different times \(t1\) and \(t2\), the heights \(s(t1)\) and \(s(t2)\) are different. Since the height of Freedom 7 varied continuously throughout its flight, the function \( s \) guarantees different heights for different times.

Step 3: Show that \(s\) is a one-to-one function

Conclude that function \(s\), which represents the height of spacecraft Freedom 7 as a function of time, is one-to-one, because for every minute during the journey, Freedom 7 was at a different height.

Key Concepts

Understanding Function Behavior

To understand the behavior of a function, think about how input values determine unique output values. A function illustrates a relationship where each input links to one specific output.
For example, in the scenario of Freedom 7, the function represents the spacecraft's height over time. The behavior of the function closely mirrors the spacecraft's movement.

• Initially, as the spacecraft ascends, the height increases over time.
• Upon reaching its peak, the height stabilizes briefly before descending.

Identifying a function as either increasing, decreasing, or constant over a period helps us understand its behavior. The key for a one-to-one function is that it must consistently produce distinct output heights for each moment in time.

Concept of Ballistic Trajectory

A ballistic trajectory describes the path followed by an object under the force of gravity once it has been propelled. This path consists of an upward movement, reaching a peak, followed by a downward path.
In the case of Freedom 7, this means:

• Initially, the spacecraft ascends, gaining height momentarily because of the initial force.
• It reaches its maximum height – the top of the trajectory.
• Finally, it begins its descent back toward the Earth.

Understanding this concept is crucial because it naturally ensures that the height changes continuously. So, for any two moments during the flight, the corresponding heights will be different. Hence, the function describing this path will be one-to-one.

Height Function of Freedom 7

The height function, denoted by \( s(t) \), represents the height of Freedom 7 over time \( t \). This function is designed to indicate how the height changes as time progresses during the flight.
Things to note about \( s(t) \):

• The maximum height, as recorded for Freedom 7, is 116 miles.
• Every point in time \( t \) during the flight maps to a single distinct height \( s(t) \).

This trait makes the function a precise representation of the spacecraft's journey, with the time variable noting different moments during the flight and the function output mapping the corresponding actual heights.

Mathematics Reasoning for One-to-One Functions

A one-to-one function, by mathematical reasoning, means that for every unique input, there is a distinct output. If \( s(t_1) = s(t_2) \), then \( t_1 \) must equal \( t_2 \). This condition is paramount in verifying that no two different times share the same height during the flight of Freedom 7.
Why is this significant?

• It confirms that at every instant during the flight, the spacecraft was at a unique height.
• By avoiding repeated height values for different times, it ensures clear and accurate tracking of the spacecraft's trajectory.

Understanding this reasoning is crucial for grasping the requirement and significance of one-to-one functions in mathematical reasoning.