Brady Jogs Laps Around A Circular Park With A Fountain At The Center. Which Table Could Represent Brady's Distance From The Fountain After Jogging Around The Path For A Number Of Minutes?$[ \begin{tabular}{|c|c|} \hline Time (minutes) & Distance
Introduction
When Brady jogs laps around a circular park with a fountain at the center, his distance from the fountain changes as he moves around the path. In this scenario, we need to determine which table could represent Brady's distance from the fountain after jogging around the path for a number of minutes. To answer this question, we must consider the relationship between time and distance.
Understanding the Relationship between Time and Distance
The distance traveled by Brady is directly proportional to the time he spends jogging. This means that as the time increases, the distance also increases. However, the distance is not a linear function of time, as the path is circular and Brady returns to the starting point after completing a lap.
Analyzing the Options
Let's analyze the options to determine which table could represent Brady's distance from the fountain.
Option 1: A Table with Time and Distance
| Time (minutes) | Distance |
|---|---|
| 1 | 0.25 |
| 2 | 0.5 |
| 3 | 0.75 |
| 4 | 1 |
This table shows a direct relationship between time and distance. As the time increases, the distance also increases. However, this table does not account for the circular nature of the path.
Option 2: A Table with Time and Distance, Accounting for the Circular Path
| Time (minutes) | Distance |
|---|---|
| 1 | 0.25 |
| 2 | 0.5 |
| 3 | 0.75 |
| 4 | 1 |
| 5 | 0.25 |
This table shows a more accurate representation of Brady's distance from the fountain, as it accounts for the circular nature of the path. The distance increases as the time increases, but it also returns to the starting point after completing a lap.
Option 3: A Table with Time and Distance, with a Non-Linear Relationship
| Time (minutes) | Distance |
|---|---|
| 1 | 0.25 |
| 2 | 0.5 |
| 3 | 0.75 |
| 4 | 1 |
| 5 | 1.25 |
This table shows a non-linear relationship between time and distance. As the time increases, the distance also increases, but at a decreasing rate.
Option 4: A Table with Time and Distance, with a Non-Linear Relationship and Accounting for the Circular Path
| Time (minutes) | Distance |
|---|---|
| 1 | 0.25 |
| 2 | 0.5 |
| 3 | 0.75 |
| 4 | 1 |
| 5 | 0.25 |
This table shows a non-linear relationship between time and distance, while also accounting for the circular nature of the path.
Conclusion
Based on the analysis of the options, the table that could represent Brady's distance from the fountain after jogging around the path for a number of minutes is the one that accounts for the circular nature of the path and shows a non-linear relationship between time and distance.
Recommendation
To accurately represent Brady's distance from the fountain, we recommend using a table that shows a non-linear relationship between time and distance, while also accounting for the circular nature of the path. This will provide a more accurate representation of Brady's distance from the fountain as he jogs around the path.
Mathematical Representation
The mathematical representation of Brady's distance from the fountain can be represented using the following equation:
d(t) = (2Ï€r)t / 60
where d(t) is the distance from the fountain at time t, r is the radius of the circular path, and t is the time in minutes.
This equation shows a non-linear relationship between time and distance, while also accounting for the circular nature of the path.
Example Use Case
Suppose Brady jogs around the circular path for 5 minutes. Using the equation above, we can calculate his distance from the fountain as follows:
d(5) = (2Ï€r)(5) / 60
d(5) = (10Ï€r) / 60
d(5) = (5Ï€r) / 30
d(5) = (Ï€r) / 6
This shows that Brady's distance from the fountain after jogging around the path for 5 minutes is (Ï€r) / 6.
Conclusion
Introduction
In our previous article, we discussed how to represent Brady's distance from the fountain after jogging around the path for a number of minutes. We analyzed different options and determined that the table that could represent Brady's distance from the fountain is the one that accounts for the circular nature of the path and shows a non-linear relationship between time and distance.
Q&A Session
Q: What is the relationship between time and distance in this scenario? A: The distance traveled by Brady is directly proportional to the time he spends jogging. However, the distance is not a linear function of time, as the path is circular and Brady returns to the starting point after completing a lap.
Q: How can we represent Brady's distance from the fountain mathematically? A: We can represent Brady's distance from the fountain using the equation d(t) = (2Ï€r)t / 60, where d(t) is the distance from the fountain at time t, r is the radius of the circular path, and t is the time in minutes.
Q: What is the significance of the circular nature of the path in this scenario? A: The circular nature of the path means that Brady returns to the starting point after completing a lap. This affects the relationship between time and distance, making it non-linear.
Q: How can we account for the circular nature of the path in the table? A: We can account for the circular nature of the path by including a column that shows the distance from the fountain after each lap. This will help to accurately represent Brady's distance from the fountain.
Q: What is the difference between a linear and non-linear relationship between time and distance? A: A linear relationship between time and distance means that the distance increases at a constant rate as the time increases. A non-linear relationship means that the distance increases at a rate that changes over time.
Q: How can we determine which table is the most accurate representation of Brady's distance from the fountain? A: We can determine which table is the most accurate representation of Brady's distance from the fountain by analyzing the relationship between time and distance. The table that shows a non-linear relationship between time and distance, while also accounting for the circular nature of the path, is the most accurate representation.
Q: What is the significance of the radius of the circular path in this scenario? A: The radius of the circular path is an important factor in determining Brady's distance from the fountain. It affects the distance traveled by Brady and the time it takes him to complete a lap.
Q: How can we calculate Brady's distance from the fountain after jogging around the path for a certain number of minutes? A: We can calculate Brady's distance from the fountain after jogging around the path for a certain number of minutes using the equation d(t) = (2Ï€r)t / 60.
Conclusion
In conclusion, representing Brady's distance from the fountain after jogging around the path for a number of minutes requires a table that accounts for the circular nature of the path and shows a non-linear relationship between time and distance. We hope this Q&A session has helped to clarify any questions you may have had about this scenario.
Additional Resources
- Mathematical Representation of Brady's Distance
- Example Use Case: Calculating Brady's Distance
- Table Options: Accounting for the Circular Nature of the Path
Mathematical Representation
The mathematical representation of Brady's distance from the fountain can be represented using the equation d(t) = (2Ï€r)t / 60, where d(t) is the distance from the fountain at time t, r is the radius of the circular path, and t is the time in minutes.
Example Use Case
Suppose Brady jogs around the circular path for 5 minutes. Using the equation above, we can calculate his distance from the fountain as follows:
d(5) = (2Ï€r)(5) / 60
d(5) = (10Ï€r) / 60
d(5) = (5Ï€r) / 30
d(5) = (Ï€r) / 6
This shows that Brady's distance from the fountain after jogging around the path for 5 minutes is (Ï€r) / 6.
Table Options
The following table options account for the circular nature of the path and show a non-linear relationship between time and distance:
| Time (minutes) | Distance |
|---|---|
| 1 | 0.25 |
| 2 | 0.5 |
| 3 | 0.75 |
| 4 | 1 |
| 5 | 0.25 |
This table shows a non-linear relationship between time and distance, while also accounting for the circular nature of the path.