## Step 1: Define the symbolic representation of the variables
The variables are ['hours worked by Laura', 'hours worked by Paul', 'hours worked by Mary', 'hours worked by Bill', 'hours worked by John']. Let's denote them as $x_0, x_1, x_2, x_3, x_4$ respectively.

## Step 2: Define the objective function in symbolic notation
The objective function to minimize is $4.18x_0 + 4.81x_1 + 7.49x_2 + 5.02x_3 + 9.16x_4$.

## Step 3: List the constraints in symbolic notation
Constraints:
1. $4.22x_0 + 6.0x_3 \geq 55$
2. $4.22x_0 + 6.81x_4 \geq 33$
3. $4.27x_2 + 6.81x_4 \geq 45$
4. $4.22x_0 + 4.27x_2 \geq 37$
5. $7.07x_1 + 4.27x_2 \geq 39$
6. $7.07x_1 + 6.0x_3 + 6.81x_4 \geq 49$
7. $4.27x_2 + 6.0x_3 + 6.81x_4 \geq 49$
8. $7.07x_1 + 4.27x_2 + 6.81x_4 \geq 49$
9. $7.07x_1 + 6.0x_3 + 6.81x_4 \geq 54$
10. $6.0x_3 + 6.81x_4 \geq 54$
11. $7.07x_1 + 4.27x_2 + 6.81x_4 \geq 58$
12. $4.22x_0 + 7.07x_1 + 4.27x_2 + 6.0x_3 + 6.81x_4 \geq 58$
13. $3.34x_0 + 8.31x_3 \geq 16$
14. $3.34x_0 + 7.32x_2 \geq 9$
15. $3.34x_0 + 5.31x_4 \geq 15$
16. $1.13x_1 + 7.32x_2 \geq 9$
17. $7.32x_2 + 8.31x_3 + 5.31x_4 \geq 14$
18. $3.34x_0 + 1.13x_1 + 7.32x_2 + 8.31x_3 + 5.31x_4 \geq 14$
19. $10x_0 - 10x_3 \geq 0$
20. $3.34x_0 + 8.31x_3 \leq 87$
21. $1.13x_1 + 7.32x_2 \leq 54$
22. $3.34x_0 + 1.13x_1 \leq 27$
23. $8.31x_3 + 5.31x_4 \leq 83$
24. $3.34x_0 + 5.31x_4 \leq 89$
25. $1.13x_1 + 8.31x_3 \leq 27$
26. $8.31x_3 + 5.31x_4 \leq 33$
27. $1.13x_1 + 5.31x_4 \leq 28$
28. $3.34x_0 + 7.32x_2 \leq 81$
29. $3.34x_0 + 8.31x_3 + 5.31x_4 \leq 90$
30. $1.13x_1 + 7.32x_2 + 8.31x_3 \leq 75$
31. $3.34x_0 + 1.13x_1 + 7.32x_2 \leq 22$

## Step 4: Determine the variable types
- $x_0$ (hours worked by Laura) can be non-integer.
- $x_1$ (hours worked by Paul) can be non-integer.
- $x_2$ (hours worked by Mary) must be an integer.
- $x_3$ (hours worked by Bill) can be non-integer.
- $x_4$ (hours worked by John) can be non-integer.

## 5: Write the Gurobi code
```python
import gurobi

# Define the model
m = gurobi.Model()

# Define the variables
x0 = m.addVar(name="x0", lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY)
x1 = m.addVar(name="x1", lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY)
x2 = m.addVar(name="x2", lb=0, ub=gurobi.GRB.INFINITY, vtype=gurobi.GRB.INTEGER)
x3 = m.addVar(name="x3", lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY)
x4 = m.addVar(name="x4", lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY)

# Define the objective function
m.setObjective(4.18*x0 + 4.81*x1 + 7.49*x2 + 5.02*x3 + 9.16*x4, gurobi.GRB.MINIMIZE)

# Add constraints
m.addConstr(4.22*x0 + 6.0*x3 >= 55)
m.addConstr(4.22*x0 + 6.81*x4 >= 33)
m.addConstr(4.27*x2 + 6.81*x4 >= 45)
m.addConstr(4.22*x0 + 4.27*x2 >= 37)
m.addConstr(7.07*x1 + 4.27*x2 >= 39)
m.addConstr(7.07*x1 + 6.0*x3 + 6.81*x4 >= 49)
m.addConstr(4.27*x2 + 6.0*x3 + 6.81*x4 >= 49)
m.addConstr(7.07*x1 + 4.27*x2 + 6.81*x4 >= 49)
m.addConstr(7.07*x1 + 6.0*x3 + 6.81*x4 >= 54)
m.addConstr(6.0*x3 + 6.81*x4 >= 54)
m.addConstr(7.07*x1 + 4.27*x2 + 6.81*x4 >= 58)
m.addConstr(4.22*x0 + 7.07*x1 + 4.27*x2 + 6.0*x3 + 6.81*x4 >= 58)
m.addConstr(3.34*x0 + 8.31*x3 >= 16)
m.addConstr(3.34*x0 + 7.32*x2 >= 9)
m.addConstr(3.34*x0 + 5.31*x4 >= 15)
m.addConstr(1.13*x1 + 7.32*x2 >= 9)
m.addConstr(7.32*x2 + 8.31*x3 + 5.31*x4 >= 14)
m.addConstr(3.34*x0 + 1.13*x1 + 7.32*x2 + 8.31*x3 + 5.31*x4 >= 14)
m.addConstr(10*x0 - 10*x3 >= 0)
m.addConstr(3.34*x0 + 8.31*x3 <= 87)
m.addConstr(1.13*x1 + 7.32*x2 <= 54)
m.addConstr(3.34*x0 + 1.13*x1 <= 27)
m.addConstr(8.31*x3 + 5.31*x4 <= 83)
m.addConstr(3.34*x0 + 5.31*x4 <= 89)
m.addConstr(1.13*x1 + 8.31*x3 <= 27)
m.addConstr(8.31*x3 + 5.31*x4 <= 33)
m.addConstr(1.13*x1 + 5.31*x4 <= 28)
m.addConstr(3.34*x0 + 7.32*x2 <= 81)
m.addConstr(3.34*x0 + 8.31*x3 + 5.31*x4 <= 90)
m.addConstr(1.13*x1 + 7.32*x2 + 8.31*x3 <= 75)
m.addConstr(3.34*x0 + 1.13*x1 + 7.32*x2 <= 22)

# Solve the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("x0: ", x0.varValue)
    print("x1: ", x1.varValue)
    print("x2: ", x2.varValue)
    print("x3: ", x3.varValue)
    print("x4: ", x4.varValue)
else:
    print("The model is infeasible")
```

## 6: Symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Laura'),
        ('x1', 'hours worked by Paul'),
        ('x2', 'hours worked by Mary'),
        ('x3', 'hours worked by Bill'),
        ('x4', 'hours worked by John')
    ],
    'objective_function': '4.18*x0 + 4.81*x1 + 7.49*x2 + 5.02*x3 + 9.16*x4',
    'constraints': [
        '4.22*x0 + 6.0*x3 >= 55',
        '4.22*x0 + 6.81*x4 >= 33',
        '4.27*x2 + 6.81*x4 >= 45',
        '4.22*x0 + 4.27*x2 >= 37',
        '7.07*x1 + 4.27*x2 >= 39',
        '7.07*x1 + 6.0*x3 + 6.81*x4 >= 49',
        '4.27*x2 + 6.0*x3 + 6.81*x4 >= 49',
        '7.07*x1 + 4.27*x2 + 6.81*x4 >= 49',
        '7.07*x1 + 6.0*x3 + 6.81*x4 >= 54',
        '6.0*x3 + 6.81*x4 >= 54',
        '7.07*x1 + 4.27*x2 + 6.81*x4 >= 58',
        '4.22*x0 + 7.07*x1 + 4.27*x2 + 6.0*x3 + 6.81*x4 >= 58',
        '3.34*x0 + 8.31*x3 >= 16',
        '3.34*x0 + 7.32*x2 >= 9',
        '3.34*x0 + 5.31*x4 >= 15',
        '1.13*x1 + 7.32*x2 >= 9',
        '7.32*x2 + 8.31*x3 + 5.31*x4 >= 14',
        '3.34*x0 + 1.13*x1 + 7.32*x2 + 8.31*x3 + 5.31*x4 >= 14',
        '10*x0 - 10*x3 >= 0',
        '3.34*x0 + 8.31*x3 <= 87',
        '1.13*x1 + 7.32*x2 <= 54',
        '3.34*x0 + 1.13*x1 <= 27',
        '8.31*x3 + 5.31*x4 <= 83',
        '3.34*x0 + 5.31*x4 <= 89',
        '1.13*x1 + 8.31*x3 <= 27',
        '8.31*x3 + 5.31*x4 <= 33',
        '1.13*x1 + 5.31*x4 <= 28',
        '3.34*x0 + 7.32*x2 <= 81',
        '3.34*x0 + 8.31*x3 + 5.31*x4 <= 90',
        '1.13*x1 + 7.32*x2 + 8.31*x3 <= 75',
        '3.34*x0 + 1.13*x1 + 7.32*x2 <= 22'
    ]
}
```