Question

Consider the following scenario: Jayne enjoys riding her bicycle through the woods. At the forest preserve, she gets on her bicycle and rides up a 2000 -foot incline in 10 minutes. She then travels down the incline in 3 minutes. The next 5000 feet is level terrain, and she covers the distance in 20 minutes. She rests for 15 minutes. Jayne then travels 10,000 feet in 30 minutes. Draw a graph of Jayne's distance traveled (in feet) as a function of time.

Short answer

Plot points at (0,0), (10,2000), (13,4000), (33,9000), (48,9000), and (78,19000), then connect with lines.

Step-by-step solution

Understand the Scenario

Jayne's journey consists of riding up an incline, down an incline, across level terrain, resting, and then traveling more level terrain. We need to plot her distance traveled over time.

Convert Time to Consistent Units

Convert all times to minutes to keep consistent units. Summarizing, Jayne spends 10 minutes going up, 3 minutes going down, 20 minutes on level terrain, rests for 15 minutes, and then rides for 30 minutes.

Calculate Cumulative Distance

Break down the distances covered in each segment. 1. Up the incline: 2000 feet in 10 minutes2. Down the incline: 2000 feet in 3 minutes 3. Across level terrain: 5000 feet in 20 minutes4. Rest: 0 feet in 15 minutes 5. Across more level terrain: 10,000 feet in 30 minutes

Create Data Points

Create data points for each segment to account for cumulative distance over time: 1. (0 min, 0 feet) - start 2. (10 min, 2000 feet) - top of the incline 3. (13 min, 4000 feet) - bottom of the incline 4. (33 min, 9000 feet) - end of first level terrain 5. (48 min, 9000 feet) - after rest 6. (78 min, 19,000 feet) - end of journey

Plot the Graph

Plot each data point from Step 4 on the graph using time (minutes) along the x-axis and cumulative distance (feet) along the y-axis. Connect the points with straight lines.- Between (0,0) and (10,2000) - upward slope- Between (10,2000) and (13,4000) - downward slope- Between (13,4000) and (33,9000) - flat line- Between (33,9000) and (48,9000) - horizontal line during rest- Between (48,9000) and (78,19000) - upward slope

Key concepts

graphing functions

Graphing functions helps us visualise how variables like distance and time interact with each other. It can show trends and allow us to predict future outcomes. For example, in Helen's biking journey, we can see how the distance Jayne travels changes over time. We use a distance-time graph to plot this relationship, where distance (in feet) is plotted on the y-axis, and time (in minutes) is plotted on the x-axis. This kind of graph starts at the origin (0,0), where Jayne begins her journey. As she moves through different terrains and rests, we mark these points on the graph, creating a visual representation of her journey.

cumulative distance

Cumulative distance is the total distance traveled from the starting point to a given point. Throughout Jayne's journey, we keep adding the distance she covers in different segments. For instance, after riding 2000 feet up an incline in 10 minutes, her cumulative distance is 2000 feet. When she rides down another 2000 feet in 3 minutes, her cumulative distance increases to 4000 feet. Calculating cumulative distance helps us plot accurate data points on our distance-time graph.

time units conversion

Time units conversion is vital to maintain consistency when analyzing data. In Jayne's biking scenario, various segments of her trip are measured in minutes. Converting all time segments to the same unit (minutes) ensures a smooth plotting process. Jayne rides 10 minutes up, 3 minutes down, 20 minutes on level terrain, rests for 15 minutes, and then rides again for 30 minutes. Summing these give us total time intervals to correctly plot the cumulative distances at each step of her journey.

plotting data points

Plotting data points accurately on a graph visualizes the relationship between the time and distance Jayne travels. Each segment from her journey provides these data points. Here's how she progresses: she starts at (0 min, 0 feet), reaches (10 min, 2000 feet), then (13 min, 4000 feet). Continuing to (33 min, 9000 feet), resting at (48 min, 9000 feet), and finally reaching her destination at (78 min, 19000 feet). By connecting these points with straight lines, we create a continuous representation of her trip on our graph. This shows periods of movement and rest, including changes in speed across inclines and flat terrain.