Question

graph the situation. You start from home and drive 55 miles per hour for 3 hours, where \(d\) is your distance from home.

Short answer

You will be 165 miles away from home after 3 hours of driving at a constant speed of 55 miles per hour. The graph is a straight line starting at the origin (0,0) and ending at the point (3, 165).

Step-by-step solution

Understanding the problem

It is known that you start from home and drive 55 miles per hour for 3 hours. The speed is constant so you need to apply the formula for distance given constant speed: 'distance = speed * time', where speed is 55 miles per hour and time is 3 hours.

Calculate the distance

Use the 'distance = speed * time' formula to calculate the distance: \(d = 55 * 3 = 165\) miles, so you will be 165 miles away from home after 3 hours of driving at a constant speed of 55 miles per hour.

Plotting the Graph

You can plot this on a graph where the x-axis represents time (in hours) and the y-axis represents distance from home (in miles). At 'time=0', the 'distance=0' which is the starting point (home). As the time increases, the distance also increases linearly. After 3 hours (at 'time=3'), the 'distance=165' miles. This will be a straight line graph starting from the origin (0,0) and ending at (3,165).

Key concepts

Distance-Speed-Time Relationship

Understanding the distance-speed-time relationship is crucial for solving many real-world problems involving motion. At the heart of this concept is a simple equation that captures the essence of how these three variables are connected:

\[ \text{Distance} = \text{Speed} \times \text{Time} \]

This equation tells us that the distance traveled is equal to the speed at which one travels multiplied by the time spent traveling. In our example, you are driving at a constant speed of 55 miles per hour for 3 hours. By applying the formula, we get:

\[ d = 55 \text{ miles per hour} \times 3 \text{ hours} = 165 \text{ miles} \]

This result indicates that you'll be 165 miles away from home after the 3-hour journey. The practical application of this formula is vast, ranging from simple road trips to complex scientific calculations.

Linear Functions

Linear functions are mathematical expressions that create straight lines when plotted on a graph. These functions are typically written in the form:

\[ y = mx + b \]

where \( y \) is the dependent variable, \( x \) is the independent variable, \( m \) is the slope of the line, and \( b \) is the y-intercept, or the point where the line crosses the y-axis. In the context of distance and speed, the equation can be thought of as:

\[ \text{Distance} = (\text{Speed}) \times (\text{Time}) + \text{Initial Distance} \]

If we assume you start from home, the initial distance is zero, and the equation simplifies to the one we have used. Here, speed corresponds to the slope (\( m \)) and time to the independent variable (\( x \)). In our example, the linear function representing your trip would be \( d = 55t \), with a slope of 55 indicating the constant speed.

Plotting Graphs

Plotting graphs is a way to visually represent data and understand how variables interact with one another. To plot a graph based on our problem:

• Choose your x-axis to represent time in hours, and your y-axis to represent distance in miles.
• Mark the starting point at (0,0), since you begin at home and hence, the initial distance is zero.
• Since you travel at a constant speed, for every hour passed, plot a point that corresponds to 55 miles further from home.
• Draw a straight line through these points to represent the constant speed.

After 3 hours, you would plot a point at (3, 165), signifying a distance of 165 miles from home. The resulting graph will show a straight line that begins at the origin and ends at this point, clearly illustrating the concept of constant speed.

Constant Speed

Constant speed means that an object is moving at the same rate throughout its journey—no speeding up, slowing down, or coming to a halt. When an object travels at a constant speed, it covers equal distances in equal intervals of time. This predictable pattern results in a linear relationship between distance and time which makes for straightforward graphing and calculations. As seen in the example, a constant speed of 55 miles per hour yields a consistent increase in distance with every hour that passes. This uniform motion is characteristic of linear functions and is represented graphically by a straight line with a slope that reflects the object's speed.