## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Hank', 'hours worked by Peggy', 'hours worked by Dale', and 'hours worked by Laura', which can be represented symbolically as $x_0$, $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Define the objective function in symbolic notation
The objective function to minimize is $3x_0 + 1(x_1 + 1) + 1x_2 + 6x_3$.

## Step 3: List the coefficients for the objective function
- $x_0$: 3
- $x_1$: 1
- $x_2$: 1
- $x_3$: 6

## 4: Define the symbolic representation of the constraints
1. $9.78x_0 + 10.71x_1 \geq 6$
2. $10.71x_1 + 0.25x_3 \geq 6$
3. $9.78x_0 + 0.25x_3 \geq 18$
4. $9.78x_0 + 1.77x_2 \geq 11$
5. $9.78x_0 + 10.71x_1 + 1.77x_2 + 0.25x_3 \geq 11$
6. $9.7x_0 + 5.48x_3 \geq 8$
7. $7.36x_1 + 6.5x_2 \geq 16$
8. $9.7x_0 + 6.5x_2 \geq 19$
9. $6.5x_2 + 5.48x_3 \geq 12$
10. $7.36x_1 + 5.48x_3 \geq 15$
11. $7.36x_1 + 6.5x_2 + 5.48x_3 \geq 9$
12. $9.7x_0 + 7.36x_1 + 6.5x_2 \geq 9$
13. $7.36x_1 + 6.5x_2 + 5.48x_3 \geq 17$
14. $9.7x_0 + 7.36x_1 + 6.5x_2 \geq 17$
15. $9.7x_0 + 7.36x_1 + 6.5x_2 + 5.48x_3 \geq 17$
16. $1.73x_0 + 7.28x_3 \geq 15$
17. $1.73x_0 + 5.19x_1 \geq 8$
18. $5.19x_1 + 7.26x_2 \geq 14$
19. $1.73x_0 + 7.26x_2 \geq 15$
20. $1.73x_0 + 5.19x_1 + 7.26x_2 + 7.28x_3 \geq 15$
21. $x_0 - 8x_2 \geq 0$
22. $10.71x_1 + 0.25x_3 \leq 28$
23. $6.5x_2 + 5.48x_3 \leq 72$
24. $9.7x_0 + 5.48x_3 \leq 58$
25. $7.36x_1 + 5.48x_3 \leq 25$
26. $9.7x_0 + 7.36x_1 \leq 39$
27. $1.73x_0 + 7.26x_2 \leq 30$

## 5: Create the Gurobi model and variables
We will now create a Gurobi model and define the variables.

## 6: Implement the objective function and constraints in Gurobi
```python
import gurobi as gp

# Create a new model
m = gp.Model("optimization_problem")

# Define the variables
x0 = m.addVar(name="x0", lb=0)  # hours worked by Hank
x1 = m.addVar(name="x1", lb=0)  # hours worked by Peggy
x2 = m.addVar(name="x2", lb=0)  # hours worked by Dale
x3 = m.addVar(name="x3", lb=0)  # hours worked by Laura

# Objective function
m.setObjective(3*x0 + x1 + x2 + 6*x3, gp.GRB.MINIMIZE)

# Constraints
m.addConstr(9.78*x0 + 10.71*x1 >= 6)
m.addConstr(10.71*x1 + 0.25*x3 >= 6)
m.addConstr(9.78*x0 + 0.25*x3 >= 18)
m.addConstr(9.78*x0 + 1.77*x2 >= 11)
m.addConstr(9.78*x0 + 10.71*x1 + 1.77*x2 + 0.25*x3 >= 11)
m.addConstr(9.7*x0 + 5.48*x3 >= 8)
m.addConstr(7.36*x1 + 6.5*x2 >= 16)
m.addConstr(9.7*x0 + 6.5*x2 >= 19)
m.addConstr(6.5*x2 + 5.48*x3 >= 12)
m.addConstr(7.36*x1 + 5.48*x3 >= 15)
m.addConstr(7.36*x1 + 6.5*x2 + 5.48*x3 >= 9)
m.addConstr(9.7*x0 + 7.36*x1 + 6.5*x2 >= 9)
m.addConstr(7.36*x1 + 6.5*x2 + 5.48*x3 >= 17)
m.addConstr(9.7*x0 + 7.36*x1 + 6.5*x2 >= 17)
m.addConstr(9.7*x0 + 7.36*x1 + 6.5*x2 + 5.48*x3 >= 17)
m.addConstr(1.73*x0 + 7.28*x3 >= 15)
m.addConstr(1.73*x0 + 5.19*x1 >= 8)
m.addConstr(5.19*x1 + 7.26*x2 >= 14)
m.addConstr(1.73*x0 + 7.26*x2 >= 15)
m.addConstr(1.73*x0 + 5.19*x1 + 7.26*x2 + 7.28*x3 >= 15)
m.addConstr(x0 - 8*x2 >= 0)
m.addConstr(10.71*x1 + 0.25*x3 <= 28)
m.addConstr(6.5*x2 + 5.48*x3 <= 72)
m.addConstr(9.7*x0 + 5.48*x3 <= 58)
m.addConstr(7.36*x1 + 5.48*x3 <= 25)
m.addConstr(9.7*x0 + 7.36*x1 <= 39)
m.addConstr(1.73*x0 + 7.26*x2 <= 30)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Hours worked by Hank: ", x0.varValue)
    print("Hours worked by Peggy: ", x1.varValue)
    print("Hours worked by Dale: ", x2.varValue)
    print("Hours worked by Laura: ", x3.varValue)
else:
    print("The model is infeasible")
```

```json
{
    'sym_variables': [
        ('x0', 'hours worked by Hank'), 
        ('x1', 'hours worked by Peggy'), 
        ('x2', 'hours worked by Dale'), 
        ('x3', 'hours worked by Laura')
    ], 
    'objective_function': '3*x0 + x1 + x2 + 6*x3', 
    'constraints': [
        '9.78*x0 + 10.71*x1 >= 6', 
        '10.71*x1 + 0.25*x3 >= 6', 
        '9.78*x0 + 0.25*x3 >= 18', 
        '9.78*x0 + 1.77*x2 >= 11', 
        '9.78*x0 + 10.71*x1 + 1.77*x2 + 0.25*x3 >= 11', 
        '9.7*x0 + 5.48*x3 >= 8', 
        '7.36*x1 + 6.5*x2 >= 16', 
        '9.7*x0 + 6.5*x2 >= 19', 
        '6.5*x2 + 5.48*x3 >= 12', 
        '7.36*x1 + 5.48*x3 >= 15', 
        '7.36*x1 + 6.5*x2 + 5.48*x3 >= 9', 
        '9.7*x0 + 7.36*x1 + 6.5*x2 >= 9', 
        '7.36*x1 + 6.5*x2 + 5.48*x3 >= 17', 
        '9.7*x0 + 7.36*x1 + 6.5*x2 >= 17', 
        '9.7*x0 + 7.36*x1 + 6.5*x2 + 5.48*x3 >= 17', 
        '1.73*x0 + 7.28*x3 >= 15', 
        '1.73*x0 + 5.19*x1 >= 8', 
        '5.19*x1 + 7.26*x2 >= 14', 
        '1.73*x0 + 7.26*x2 >= 15', 
        '1.73*x0 + 5.19*x1 + 7.26*x2 + 7.28*x3 >= 15', 
        'x0 - 8*x2 >= 0', 
        '10.71*x1 + 0.25*x3 <= 28', 
        '6.5*x2 + 5.48*x3 <= 72', 
        '9.7*x0 + 5.48*x3 <= 58', 
        '7.36*x1 + 5.48*x3 <= 25', 
        '9.7*x0 + 7.36*x1 <= 39', 
        '1.73*x0 + 7.26*x2 <= 30'
    ]
}
```