- Alisa's practice time is calculated as \(A = 5 \times 48 = 240\) minutes.
- The total practice time is calculated as \(T = 48 + 240 = 288\) minutes.
- The equation modeling the total time is \(T = 6J\), where \(J = 48\).
- Alisa's time is \(\boxed{240}\) minutes, and the total time is \(\boxed{288}\) minutes.
Explanation
- Analyzing the Problem Data
James practiced piano for \(48\) minutes. Alisa practiced for \(5\) times as long as James. We need to find two values: Alisa's total practice time and the combined time James and Alisa spent practicing. We also need an equation to represent the relationship between these times. Let \(J\) be James's practice time, so \(J = 48\). Let \(A\) be Alisa's practice time, which is given by the relation \(A = 5 \times J\). The total practice time \(T\) is the sum of their individual times, \(T = J + A\).
- Calculating Alisa's Practice Time
To find Alisa's practice time \(A\), we multiply James's practice time by \(5\):
\[A = 5 \times J = 5 \times 48 = 240\]
Alisa practiced for \(240\) minutes.
- Calculating Total Practice Time
To find the combined practice time \(T\), we add James's time and Alisa's time together:
\[T = J + A = 48 + 240 = 288\]
James and Alisa practiced for a total of \(288\) minutes.
- Modeling with an Equation and Final Conclusion
To model the work, we can express the total time \(T\) in terms of James's time \(J\). Since \(T = J + A\) and \(A = 5J\), we have:
\[T = J + 5J = 6J\]
Substituting \(J = 48\), we get \(T = 6 \times 48 = 288\). The equation modeling the total time is \(T = 6J\), where \(J=48\). Alisa's time is \(240\) minutes, and the total time is \(288\) minutes.
Examples
Modeling with equations is a vital skill in daily life, especially when planning tasks or managing resources. For instance, if you are organizing a team project and one member completes a task in a certain amount of time, you can estimate the total time for the team by using an equation. If person A takes \(T\) hours and person B takes \(3T\) hours, the total time is simply \(4T\). By defining these relationships, you can accurately predict project timelines and allocate your resources effectively.