Problem 61 - College Algebra with Modeling and Visualization

Question

Sketch a graph that illustrates the motion of the person described. Let the \(x\) -axis represent time and the \(y\) axis represent distance from home. Be sure to label each axis. A person drives a car away from home for 2 hours at 50 miles per hour and then stops for 1 hour.

Step-by-Step Solution

Answer

A linear increase from 0 to 100 miles over 2 hours, then a flat line at 100 miles for 1 hour.

Step 1: Understand the Problem

A person drives at a constant speed of 50 miles per hour for 2 hours and then stops for 1 hour. We are asked to sketch a graph that represents this motion.

Step 2: Setup Axes

On a coordinate plane, label the x-axis as 'Time (hours)' and the y-axis as 'Distance from Home (miles)'. Both axes should start from zero since initially, the person hasn't moved from home or spent any time.

Step 3: Plot the Driving Phase

From hour 0 to hour 2 on the x-axis, the car drives at 50 miles per hour. To find the distance at each point, multiply the time by the speed. So at 1 hour, the distance is 50 miles, and at 2 hours, the distance is 100 miles. Draw a straight line from (0,0) to (2,100) to illustrate the constant speed.

Step 4: Plot the Resting Phase

From hour 2 to hour 3 on the x-axis, the person stops. During this time, the distance does not change. Hence, draw a horizontal line from point (2,100) to (3,100) to represent the stop for 1 hour.

Step 5: Finalize the Graph

Review the graph to ensure it accurately depicts the scenario: A linear increase from (0,0) to (2,100) for the driving phase and a horizontal line from (2,100) to (3,100) for the stop. Label the points where the car stops and the start and end points of the drive.

Key Concepts

Coordinate Plane

The coordinate plane is a fundamental concept in graphing motion. Think of it as a map showing how one thing changes in relation to another. In this task, the coordinate plane helps us explore how time affects distance.
The axes are like guideposts:

• The horizontal line is called the x-axis. Here, it represents time (in hours).
• The vertical line is the y-axis, which shows distance (in miles) from home.

We start both axes at zero because at the beginning of the journey, neither time nor distance has accumulated. As time progresses, the changes in distance are marked on the graph, illustrating the person's movement over time.

Distance-Time Graph

A distance-time graph visually represents the relationship between distance and time. It displays how far someone or something travels over a specific time period.
To create this graph, locate the different events on your coordinate plane:

• During the driving phase, the person covers 50 miles every hour. This part of the graph is a straight line indicating constant speed.
• The straight line starts from the origin, where neither distance nor time has passed. It goes up to the point (2,100), as the person travels 100 miles in 2 hours.

After 2 hours, the person's motion pauses for an hour. On the graph, this appears as a horizontal line from (2,100) to (3,100) because the distance does not change during the pause.
With these sections/directions plotted, the graph becomes a clear snapshot of the journey.

Constant Speed

Constant speed is when an object moves at the same pace, with equal distances covered in equal intervals of time. In the exercise, the person drives the car at a steady pace of 50 miles per hour.
Here's what constant speed looks like on the graph:

• The line is straight and diagonal from the point (0,0) to (2,100).
• Every hour, the distance increases by 50 miles. It's predictable and uniform.

Constant speed leads to a constant slope on the graph. A steeper slope would imply a faster speed, but here it remains consistent at 50 miles per hour during the first 2 hours. Understanding constant speed helps make sense of how the journey progresses smoothly before the stop.