A project manager came up with the following network diagram. (i) Calculate the earliest starting time of activity F. (ii) Calculate the duration of the project.

$$\text{EET(Node 1)} = 0 \text{ days}$$ $$\text{EET(Node 2)} = \text{EET(Node 1)} + \text{Duration(A)} = 0 + 5 = 5 \text{ days}$$ $$\text{EET(Node 3)} = \text{EET(Node 1)} + \text{Duration(B)} = 0 + 3 = 3 \text{ days}$$

Node 4 is preceded by activities C (from Node 2) and D (from Node 3).

$$\text{EET(Node 4)} = \max(\text{EET(Node 2)} + \text{Duration(C)}, \text{EET(Node 3)} + \text{Duration(D)})$$ $$\text{EET(Node 4)} = \max(5 + 4, 3 + 6)$$ $$\text{EET(Node 4)} = \max(9, 9) = 9 \text{ days}$$

The earliest starting time of activity F is the earliest event time of its preceding node, which is Node 4.
The earliest starting time of activity F is $\boxed{\text{9 days}}$.

Node 5 is preceded by activities F (from Node 4) and E (from Node 3).

$$\text{EET(Node 5)} = \max(\text{EET(Node 4)} + \text{Duration(F)}, \text{EET(Node 3)} + \text{Duration(E)})$$ $$\text{EET(Node 5)} = \max(9 + 8, 3 + 2)$$ $$\text{EET(Node 5)} = \max(17, 5) = 17 \text{ days}$$

The duration of the project is the earliest event time of the final node.
The duration of the project is $\boxed{\text{17 days}}$.