Problem 44 - Contemporary Precalculus

Question

A plane flies from Austin, Texas, to Cleveland, Ohio, a distance of 1200 miles. Let \(f\) be the function whose rule is \(f(t)=\) distance (in miles) from Austin at time \(t\) hours. Draw a plausible graph of \(f\) under the given circumstances. IThere are many possible correct answers for each part. \(]\) (a) The flight is nonstop and takes less than 4 hours. (b) Bad weather forces the plane to land in Dallas (about 200 miles from Austin), remain overnight (for 8 hours), and continue the next day. (c) The flight is nonstop, but owing to heavy traffic, the plane must fly in a holding pattern over Cincinnati (about 200 miles from Cleveland) for an hour before going on to Cleveland.

Step-by-Step Solution

Answer

Question: Describe the main characteristics of the distance-time graphs for each of the following situations: A) Nonstop flight under 4 hours, B) Plane stops in Dallas, and C) Holding pattern over Cincinnati. Answer: A) For a nonstop flight under 4 hours, the graph will show a continuous straight line with a constant slope, representing a constant speed. B) For a plane stopping in Dallas, the graph will show an increase in distance to 200 miles, remain constant at 200 miles for 8 hours, and then continue increasing to 1200 miles. C) For a holding pattern over Cincinnati, the graph will show a continuous increase to 1000 miles, remain constant at 1000 miles for 1 hour, and then continue increasing to 1200 miles.

Step 1: Situation A: Nonstop flight under 4 hours

For this situation, the graph should show a continuous increase in distance from 0 miles (Austin) to 1200 miles (Cleveland). The time of the flight should be less than 4 hours. Since the flight is nonstop, the graph should be a straight line with a constant slope. The slope represents the plane's speed and can be found by dividing the total distance by the time.

Step 2: Situation B: Plane stops in Dallas

In this situation, the plane first travels 200 miles to Dallas, where it stays for 8 hours before continuing the flight. The graph should show a continuous increase in distance from 0 miles to 200 miles (Dallas), where it would then remain at 200 miles for 8 hours, indicating no change in distance during that time. After the 8-hour stop, the graph should show another continuous increase in distance from 200 miles to 1200 miles (Cleveland).

Step 3: Situation C: Holding pattern over Cincinnati

In this case, the flight is nonstop, but the plane has to fly in a holding pattern over Cincinnati for 1 hour before going to Cleveland. The graph should show a continuous increase in distance from 0 miles (Austin) to 1000 miles (Cincinnati). At this point, the plane would fly in a holding pattern, maintaining a constant distance from Austin. The graph should show a horizontal line at 1000 miles for 1 hour, indicating no change in distance. After the holding pattern, the graph should continue with a continuous increase in distance from 1000 miles to 1200 miles (Cleveland).

Key Concepts

Piecewise Functions

Piecewise functions are a great way to describe different behaviors within the same function. They are like instructions with different steps that depend on the input value. This type of function breaks down into sections, or "pieces," each with its own rule.
For instance, when discussing the plane's flight from Austin to Cleveland, the function describing the plane's distance can change based on events such as stops or holding patterns.

• Consider a function for a nonstop flight that describes a constant rate or speed from start to destination without interruption. This is a single piece of the function.
• However, if the plane stops in Dallas, the function has multiple pieces: one for the flight to Dallas, a constant piece while waiting, and another for continuing to Cleveland.
• This way of structuring a function allows us to model real-life scenarios that don't fit neatly into one pattern.

By visualizing and creating piecewise functions, you efficiently demonstrate the sequence of events over a set timeframe through changes in the slope and pattern of the graph.

Linear Functions

Linear functions are the simplest kind of functions often used to model relationships with a constant rate of change, which appears as a straight line on a graph. The equation of a linear function can be written as: \( y = mx + b \), where \( m \) is the slope of the line, indicating how steep it is, and \( b \) is the y-intercept, where the line crosses the y-axis.
In the context of the flight, when the flight is nonstop and under constant speed, the graph of the distance function is a straight line. The slope \( m \)in this scenario represents the speed of the plane, calculated as \( \text{distance traveled} / \text{time} \). Additionally, \( b \) is \( 0 \) because the plane starts at 0 miles from Austin.

• During a nonstop flight to Cleveland that takes less than 4 hours, the graph shows a straight line from 0 to 1200 miles.
• The steeper the line, the faster the plane is traveling.
• If conditions like traffic or a stop over occurs, we introduce another pattern and thus move to our previous discussion on piecewise functions.

Understanding the properties of linear functions aids in interpreting and predicting distances and times efficiently.

Distance-Time Graphs

Distance-time graphs are a helpful tool used to visualize how far something has moved over time. These graphs plot time on the x-axis and distance on the y-axis. They show not only the distance covered but also the speed of travel.
Such graphs are especially insightful when reflecting on our plane flight scenarios:

• A straight, upward-sloping line represents constant speed (like the nonstop flight), indicating steady progress in distance over time.
• A horizontal line demonstrates zero speed, where the distance remains constant; for example, when the plane waits in Dallas overnight.
• Changes in the slope of the line will show changes in speed or direction, such as during the holding pattern over Cincinnati.

The gradient or slope of the line is directly linked to speed. A steeper slope correlates with a higher speed, while a flatter slope shows slower movement. Understanding how to read and interpret distance-time graphs allows us to make sense of travel patterns and time spent moving or stationary.