Question

You are supervising the construction of a roller coaster for young children. For the first 20 feet of horizontal distance, the track must rise off the ground at a constant rate. After your crew has constructed 5 feet of horizontal distance, the track is 1 foot off the ground. Plot points for the heights of the track at 5 -foot intervals. Draw a line through the points. Find the slope of the line. What does the slope represent?

Short answer

The slope of the line is 0.2, which represents the rate at which the height of the roller coaster increases with respect to the horizontal distance covered. Therefore, for each foot of horizontal distance covered, the track rises by 0.2 feet.

Step-by-step solution

Plot the points

Given the information, we can identify the point (5,1) where at 5 feet of horizontal distance, the track is 1 foot high. Since the track rises at a constant rate, every 5 feet of horizontal distance, the track will rise equally. Therefore we can derive other points: (0,0), (10,2), (15,3) and (20,4). Plot these points on a graph, with the horizontal axis representing the distance covered and the vertical axis representing the height of the track.

Draw the line

Through the plotted points, draw a line that represents the rising track of the roller coaster. This line will be straight, indicating a constant climb.

Calculate the slope

The slope of a line is given by the ratio of the vertical change (rise) to the horizontal change (run). With any two given points for instance, (5,1) and (10,2), we can calculate the slope, m, using the formula: m= (y2-y1)/(x2-x1), where (x1,y1) are the coordinates of the first point and (x2,y2) are the coordinates of the second point. Substituting the given values, we get m=(2-1)/(10-5) = 1/5 = 0.2.

Interpret the slope

The slope in this context signifies the rate at which the height of the roller coaster increases with respect to the horizontal distance covered. Therefore, a slope of 0.2 indicates that for each foot of horizontal distance covered, the track rises by 0.2 feet.

Key concepts

Coordinate Graphing

Understanding coordinate graphing is crucial not just in algebra, but in a variety of real-life applications, like designing a child-safe roller coaster as in our scenario. In coordinate graphing, we use two perpendicular number lines: the horizontal axis, often dubbed the x-axis, and the vertical axis, called the y-axis. By marking a point where these lines intersect, we are pinpointing an exact location on the plane, given as a pair of numbers (x, y).

In our roller coaster construction exercise, we chart the rise of the track at several points along its horizontal development. We map these as coordinates: for example, (5, 1) tells us that at 5 feet of horizontal length, the roller coaster is 1 foot off the ground. It's much like plotting waypoints on a treasure map, each point guiding us to the shape and slope of our roller coaster's trajectory. With each plotted point, a clearer picture of the intended path emerges, showing how coordinate graphing translates a word problem into a visual exploration.

Linear Equations

Imagine the path of our roller coaster track as a clear, straight line, unfurling through the joyous cheers of children. This is the essence of understanding linear equations. A linear equation represents a straight line on a graph and is usually written in the form y = mx + b, where m is the slope of the line, and b is the y-intercept—the point where the line crosses the y-axis.

In our quaint roller coaster situation, the equation of the line can be derived from the points we plotted on the graph. If we connect these points linearly, we identify our track's path in the world of math. The next step is then to use the formula to compute the slope, thus leading us to see how speedily our track ascends as it snakes its way horizontally. Without linear equations, the journey from an engineer's blueprint to the towering loops of a roller coaster would lack the mathematical magic that assures safety and skeletal precision.

Rate of Change

At the heart of any good roller coaster design is the rate of change. This is not only about thrills and spills but the fundamental concept that describes how one quantity changes in relation to another. In our context, the rate of change is visually represented by the slope of the line on the graph. It's the engine behind the line's angle, dictating just how swiftly our roller coaster climbs for each foot of horizontal spread.

The formula for slope, m = (y2-y1)/(x2-x1), is our mathematical compass. For our fun-sized roller coaster exercise, the rate at which the track ascends off the ground is calculated as a steady 0.2 feet of rise for every horizontal foot. It's this gentle incline that will delight kids without overwhelming them, showcasing how the seemingly abstract concept of rate of change influences real-world decisions and designs. Most importantly, conveying the idea of rate of change to students solidifies their grasp on how variables interact—a concept as thrilling in its predictability as a safe, well-designed roller coaster.