- The rate of work is defined as \(\text{Work} / \text{Time}\).
- Rocco's work contribution in 3 hours is \(\frac{3}{7}\) of the room.
- Giulia's work contribution in 3 hours is \(3r\) of the room.
- Summing these contributions to equal the total work (1) yields the equation: \(\boxed{\frac{3}{7} + 3r = 1}\).
Explanation
- Analysis and Equation Formulation
The problem asks for an equation to determine Giulia's rate of work, denoted by \(r\), given the work done by Rocco and Giulia.
We are given the following information:
- The total amount of work to be completed is painting one room, which we define as 1 unit of work.
- Working together, Rocco and Giulia take 3 hours to complete the room.
- Working alone, it takes Rocco 7 hours to complete the room, which means his rate of work is \(\frac{1}{7}\) room per hour.
- Let Giulia's rate of work be \(r\) rooms per hour.
We can calculate the work done by each person over the 3-hour period by multiplying their respective rates by the time taken:
- Work done by Rocco = \(\text{Rate} \times \text{Time} = \frac{1}{7} \times 3 = \frac{3}{7}\).
- Work done by Giulia = \(\text{Rate} \times \text{Time} = r \times 3 = 3r\).
Since the two of them working together complete exactly one full room in 3 hours, the sum of the parts of the room they paint must equal the total work completed (1). Therefore, we add their individual contributions together:
\[\frac{3}{7} + 3r = 1\]
This equation represents the relationship between the work done by Rocco, the work done by Giulia, and the total work completed. Comparing this to the given options, it matches the second equation provided.
- Conclusion
Based on the analysis, the equation \(\frac{3}{7} + 3r = 1\) correctly captures the relationship between the individuals' work contributions and the total task completed. Thus, this is the equation that can be used to determine Giulia's rate of work.
Final answer: \(\boxed{\frac{3}{7} + 3r = 1}\).
Examples
Understanding how to set up equations for work rates is a fundamental skill in project management and personal productivity. For example, if you and a friend are cleaning your apartment, knowing your individual cleaning speeds allows you to estimate exactly how much time you need to allocate to complete the job together or to plan individual responsibilities. This same principle is used in engineering projects to determine the number of personnel or machines needed to meet a specific project deadline by balancing work rates against total work requirements.