Charting Thomas's Journey: A Distance-Time Graph Guide
Hey guys! Let's dive into a classic math problem – visualizing a road trip with a distance-time graph. We're going to chart Thomas's journey, from the moment he hits the road to when he's back home. This is a great example of how graphs can tell a story about movement, speed, and time. So, buckle up, grab your graph paper (or digital tools!), and let's get started. This journey will not only help you understand distance-time graphs, but also improve your problem-solving skills in a fun, engaging way. We'll break down each stage of Thomas's trip, calculating distances and times, and then plot everything on our graph. By the end, you'll be a distance-time graph pro!
Setting the Scene: Thomas's Grand Adventure
So, here's the scoop: Thomas leaves home at 2:00 PM. He's got a mission, and a car, and he's ready to roll. He cruises along at a steady 40 mph for a solid 3.5 hours. This part of the journey is all about covering ground at a consistent pace. But it's not all driving, folks! After those 3.5 hours, Thomas takes a 30-minute break. Everyone needs a pit stop, right? Finally, with a refreshed spirit (and probably a full tank of gas!), Thomas heads back home, this time at a zippier 70 mph. Our task is to create a distance-time graph that visually represents this entire adventure. This graph will not only illustrate the distances Thomas covered but also how the time changed. It will give a simple illustration of the story of his trip. Ready to get started?
Before we start, let's make sure we're all on the same page. A distance-time graph is a tool that shows how far an object (in this case, Thomas) has traveled over a certain amount of time. The horizontal axis (x-axis) always represents time, and the vertical axis (y-axis) represents distance. The slope of the line on the graph tells us about the object's speed. A steeper line means a faster speed. A flat line indicates that the object is not moving. If the line goes downwards, it means the object is going back towards its starting point.
Breaking Down the Journey
Let's take a closer look at each segment of Thomas's journey. This will help us understand how to calculate each part of the journey. Understanding each part will allow us to get all the data to plot into a graph.
1. Outward Journey: Thomas drives for 3.5 hours at 40 mph. How far does he go?
To figure this out, we use the formula: distance = speed Ă— time.
Distance = 40 mph Ă— 3.5 hours = 140 miles.
So, Thomas travels 140 miles in the first part of his trip.
2. The Break: Thomas stops for 30 minutes (0.5 hours). During this time, his distance doesn't change. He's just chilling.
3. Return Journey: Thomas drives home at 70 mph. We know the distance is 140 miles (the same distance as the outward journey). How long does it take him to get back?
Using the formula: time = distance / speed.
Time = 140 miles / 70 mph = 2 hours.
The return trip takes Thomas 2 hours.
Now we have all the info we need! We know the distance for each part and the time for each part. We are now ready to plot these values into the graph. Let's see how!
Plotting the Adventure: Creating the Distance-Time Graph
Alright, now for the fun part! It's time to turn our calculations into a visual story. This is where our distance-time graph comes to life. We'll create a graph to show the distances covered and the time spent on each part of the journey. Make sure you know the proper values from our previous calculations.
1. Setting up the Axes: First, draw your axes. The x-axis is for time (in hours), and the y-axis is for distance (in miles). Label them clearly. Choose appropriate scales for each axis. Make sure your graph has enough range to show the entire journey of Thomas. Remember, the distance reaches 140 miles. So, your y-axis will need to go at least that high. For the x-axis, we need to consider the whole duration of the trip. So, the graph will need to show the total time. The trip out is 3.5 hours. The break is 0.5 hours, and the trip back is 2 hours. Total trip duration is 6 hours. Thus, the x-axis should go at least that high.
2. Plotting the Outward Journey: At 2:00 PM (time 0 on our graph), Thomas starts. After 3.5 hours (at 5:30 PM), he has traveled 140 miles. So, plot a point at the coordinates (3.5, 140). Connect this point to the origin (0,0) with a straight line. The slope of this line represents Thomas's speed of 40 mph.
3. Representing the Break: For the next 30 minutes (0.5 hours), Thomas is not moving. On the graph, this is shown as a horizontal line. From 5:30 PM to 6:00 PM, he's at the same distance of 140 miles. Draw a horizontal line from the point (3.5, 140) to the point (4, 140).
4. Plotting the Return Journey: Thomas starts heading home at 6:00 PM (4 hours into his journey) and covers the 140 miles in 2 hours. Plot a point at the coordinates (6, 0) and connect the previous point with a straight line. This line will have a steeper slope than the outward journey because Thomas is traveling faster. The downward slope indicates that he's moving towards his starting point. Note the steeper angle indicating a faster speed. Remember the formula distance = speed Ă— time, the higher the speed, the steeper the angle in the graph.
5. Finishing Touches: Label the lines clearly. Add a title to your graph (e.g.,
