To find the correct equation involving Giulia's rate of work [tex]\( r \)[/tex] in parts per hour, let's break down the problem step-by-step.
1. Rocco's Rate of Work:
- Rocco can paint the room alone in 7 hours.
- Therefore, Rocco's rate of work is [tex]\(\frac{1}{7}\)[/tex] parts per hour.
2. Time Taken Together:
- Rocco and Giulia, working together, can paint the room in 3 hours.
3. Contribution of Work when Working Together:
- Together they paint one entire room (which can be considered as 1 part) in 3 hours.
- Thus, the combined rate of Rocco and Giulia is [tex]\(\frac{1}{3}\)[/tex] rooms per hour.
4. Rocco's Contribution:
- Rocco's rate is [tex]\(\frac{1}{7}\)[/tex] parts per hour, so in 3 hours, he paints [tex]\(3 \times \frac{1}{7} = \frac{3}{7}\)[/tex] of the room.
5. Giulia's Contribution:
- Let [tex]\( r \)[/tex] be Giulia's rate of work in parts per hour.
- In 3 hours, Giulia would paint [tex]\( 3r \)[/tex] parts of the room.
6. Total Work Done:
- The total work done by Rocco and Giulia together in 3 hours should be 1 entire room.
- Therefore, [tex]\( \frac{3}{7} + 3r = 1 \)[/tex].
7. Equation for Giulia's Rate of Work:
- Hence, the equation we can use to determine [tex]\( r \)[/tex] is:
[tex]\[
\frac{3}{7} + 3r = 1
\][/tex]
Thus, the equation to determine [tex]\( r \)[/tex], Giulia's rate of work in parts per hour, is:
[tex]\[
\boxed{\frac{3}{7} + 3r = 1}
\][/tex]
