Question

Modeling Data The table shows the rate \(r\) (in miles per hour) that a vehicle is traveling after \(t\) seconds. $$ \begin{array}{|c|c|c|c|c|c|c|} \hline t & 5 & 10 & 15 & 20 & 25 & 30 \\ \hline r & 57 & 74 & 85 & 84 & 61 & 43 \\ \hline \end{array} $$ (a) Plot the data by hand and connect adjacent points with a line segment. (b) Use the slope of each line segment to determine the interval when the vehicle's rate changed most rapidly. How did the rate change?

Short answer

The vehicle's rate changed most rapidly between the time intervals with the greatest positive or negative slopes.

Step-by-step solution

Creating the Plot

First, we need to plot our given data. Create a two-dimensional plot where the x-axis represents time \(t\) and the y-axis represents rate \(r\). Plot points for each corresponding pair of values, then connect each adjacent pair of plotted points with a straight line.

Calculating Slopes

Next we need to calculate the slope between each pair of adjacent points. The slope of a line segment between two points \((x_1,y_1)\) and \((x_2,y_2)\) is given by the formula \((y_2-y_1)/(x_2-x_1)\). For example, the slope between the first two points (5, 57) and (10, 74) is \((74 - 57) / (10 - 5) = 17/5 = 3.4\). Repeat this process for all adjacent point pairs.

Identifying the Rapid Changes

To find the interval(s) when the vehicle's rate changed most rapidly, look for the largest slope(s) calculated in Step 2. The corresponding time interval to this slope is the interval you are looking for. If the slope is positive, it means the rate increased during that interval. If the slope is negative, the rate decreased.

Key concepts

Modeling Data with Graphs

Translating numerical data into a visual format such as a graph can be an incredibly effective way to analyze and communicate information. The graph serves as a model of the data, making patterns easier to discern and relationships more apparent. For instance, plotting the rates of a car after certain time intervals offers a clear visual picture of how speed varies over time.

To create such a graph, each pair of values in the given data set, representing specific points in time and rates of speed, would be plotted on a two-dimensional coordinate plane, with time on the horizontal (x-axis) and rate on the vertical (y-axis). In educational exercises, students are often instructed to draw these points on graph paper and connect them with line segments to visualize the change over time.

This approach not only aids in understanding the given data but also provides a foundational technique for analyzing trends. Graphs are vital tools in various fields, including economics, science, and engineering, to model and predict behaviors.

Rate of Change

The concept of rate of change is essential in understanding how two quantities are related and how they vary concerning each other over time. In mathematical terms, this is often identified by the slope of the line segment connecting two points on a graph.

For example, the rate of change between two points on our graph can be calculated using the formula: \(\frac{r_2 - r_1}{t_2 - t_1}\), where \(r_1\) and \(r_2\) are the rates at times \(t_1\) and \(t_2\). This formula gives a numerical value expressing the rate's increase or decrease per unit time. A higher absolute value of the slope indicates a more rapid change.

Applying the Rate of Change to Real Situations

Students can deepen their understanding by exploring real-world scenarios. For instance, comprehending the rate of change in a vehicle's speed can help in optimizing fuel consumption or estimating travel times. Similarly, it provides insights into how variables change in scientific experiments or financial investments, showcasing the vast applicability of this mathematical concept.

Interpreting Slopes in Context

The slope of a line segment tells us much more than just the steepness of a line; it helps us interpret the relationship between variables in a given context. In our exercise, the slope represents how quickly a vehicle's speed changes over time.

If the slope is positive, it signifies that the vehicle is accelerating; if negative, the vehicle is decelerating. Moreover, the magnitude of the slope offers quantitative insights into the rapidity of this change. For instance, a steep slope (either positive or negative) suggests a significant change in speed over a short time interval.

Real-World Implications of Slope

Understanding slope in a real-world context is invaluable. Architects use it to design roofs, while geographers use it to assess land incline. In finance, the slope helps evaluate the trend of stock prices, and in automotive design, it informs engineers about vehicle performance under different acceleration rates. Thus, interpreting slopes within their respective contexts is a crucial skill across various professional fields, emphasizing the real-life significance of this mathematical concept.