To solve the given problem, we first need to define the decision variables and the objective function. Let's denote:

- \(x_f\) as the number of full-time employees.
- \(x_p\) as the number of part-time employees.

The objective is to minimize the total weekly wage bill. The cost of a full-time employee is $800 per week, and the cost of a part-time employee is $400 per week. Therefore, the objective function can be written as:

\[ \text{Minimize} \quad 800x_f + 400x_p \]

The constraints are as follows:

1. The total number of employees should be at least 60:
\[ x_f + x_p \geq 60 \]

2. At least 40 employees must be full-time:
\[ x_f \geq 40 \]

3. The number of full-time employees should be at least a third of the number of part-time employees:
\[ x_f \geq \frac{1}{3}x_p \]

4. The total weekly wage bill should not exceed $30,000:
\[ 800x_f + 400x_p \leq 30000 \]

Now, let's translate these constraints and the objective function into Gurobi code in Python:

```python
from gurobipy import *

# Create a new model
m = Model("Iota_Software_Optimization")

# Define the decision variables
x_f = m.addVar(vtype=GRB.CONTINUOUS, name="full_time_employees")
x_p = m.addVar(vtype=GRB.CONTINUOUS, name="part_time_employees")

# Set the objective function
m.setObjective(800*x_f + 400*x_p, GRB.MINIMIZE)

# Add constraints
m.addConstr(x_f + x_p >= 60, "total_employees")
m.addConstr(x_f >= 40, "full_time_minimum")
m.addConstr(x_f >= (1/3)*x_p, "full_time_part_time_ratio")
m.addConstr(800*x_f + 400*x_p <= 30000, "wage_bill_limit")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print("Full-time employees:", x_f.x)
    print("Part-time employees:", x_p.x)
    print("Total wage bill: $", 800*x_f.x + 400*x_p.x)
else:
    print("No optimal solution found. Status:", m.status)

```