Problem 60 - A Graphical Approach to College Algebra

Question

Sketch a graph showing the distance a person is from home after \(x\) hours if he or she drives on a straight road at 40 mph to a park 20 miles away, remains at the park for 2 hours, and then returns home at a speed of \(20 \mathrm{mph}\).

Step-by-Step Solution

Answer

The graph is a triangle: ascending to 20 miles in 0.5 hours, constant for 2 hours, then descending back to zero in 1 hour.

Step 1: Understanding the Problem

The person drives to a park, waits, and then returns home. We must graph the distance from home over time, considering the speeds and times given.

Step 2: Determine the Phases

There are three phases: (1) Driving to the park at 40 mph, (2) Staying at the park for 2 hours, and (3) Returning home at 20 mph.

Step 3: Driving to the Park

The park is 20 miles away, and the person drives at 40 mph. The time taken to reach the park is \(\frac{20}{40} = 0.5\) hours. The graph will start at 0 and increase linearly to 20 miles over 0.5 hours.

Step 4: Staying at the Park

For 2 hours, the person remains at the park 20 miles away. During this time, the distance from home is constant at 20 miles. On the graph, this appears as a horizontal line at 20 miles from 0.5 hours to 2.5 hours.

Step 5: Returning Home

The return trip is at 20 mph, covering 20 miles. The time taken is \(\frac{20}{20} = 1\) hour. From 2.5 to 3.5 hours, the graph will linearly decrease from 20 miles to 0 miles as the person returns home.

Step 6: Sketching the Graph

Combine the information into a graph: plot the points (0,0), (0.5,20), horizontal line from (0.5,20) to (2.5,20), and linearly decrease to (3.5,0). Ensure axes are labeled with time (hours) and distance (miles).

Key Concepts

Distance-Time Graph

A distance-time graph is a visual representation of the movement of an object over time. This type of graph helps to easily determine how far the object is from a starting point at different times. For our exercise, we are evaluating the journey made by a person driving to and from a park.
A distance-time graph will show distance on the vertical axis and time on the horizontal axis. Here's how to interpret the segments:

• If the line slopes upwards, the object is moving away from the starting point.
• If the line is horizontal, the object is stationary.
• If the line slopes downwards, the object is returning to the starting point.

This graph not only helps to visualize the movement but also to understand how different speeds and stops affect distance over time.

Piecewise Functions

Piecewise functions are a type of function defined by multiple sub-functions, each applying to a certain part of a function's domain. In simpler terms, this means that a different formula or rule is used for different sections of the input values.
In the exercise, the distance from home during the whole trip can be considered as a piecewise function. It consists of three distinct parts based on the actions taken:

• The trip to the park: modeled as a linear function increasing from 0 miles to 20 miles.
• The stay at the park: which is a constant function showing no change in distance for 2 hours.
• The return trip: modeled as a linear function decreasing back to 0.

Each segment of the journey uses a different part of the function, illustrating how piecewise functions can describe real-world situations by combining different models for different conditions.

Rate of Change

The rate of change measures how a quantity changes in relation to another quantity. In our distance-time graph, the rate of change is associated with speed - how quickly the distance changes over time.
For this exercise, the person's movement is divided into phases, each with a distinct rate of change:

• The initial drive to the park at 40 mph: Here, the rate is \(40 \, \text{miles per hour}\), which is seen as a steep slope on the graph.
• Staying at the park: The rate of change is 0, and the graph shows this as a flat horizontal line.
• The return journey at 20 mph: The rate is \(20 \, \text{miles per hour}\), represented by a less steep downward slope on the graph.

Understanding rate of change is crucial in interpreting how fast or slow the person is moving during different parts of the trip.