## Step 1: Define the variables and their corresponding natural language objects
The variables are ['hours worked by Hank', 'hours worked by Paul', 'hours worked by Mary', 'hours worked by Ringo', 'hours worked by Jean', 'hours worked by John']. Let's denote them as $x_0, x_1, x_2, x_3, x_4, x_5$ respectively.

## Step 2: Convert the problem description into a symbolic representation
The symbolic representation of the variables is:
- $x_0$ : hours worked by Hank
- $x_1$ : hours worked by Paul
- $x_2$ : hours worked by Mary
- $x_3$ : hours worked by Ringo
- $x_4$ : hours worked by Jean
- $x_5$ : hours worked by John

## 3: Define the objective function in symbolic notation
The objective function to minimize is: $1.73x_0 + 1.02x_1 + 2.71x_2 + 3.81x_3 + 9.75x_4 + 3.15x_5$

## 4: List the constraints in symbolic notation
Constraints:
1. $17x_1 + 8x_3 \geq 32$
2. $5x_0 + 17x_1 \geq 22$
3. $8x_3 + 7x_4 \geq 18$
4. $5x_0 + 8x_3 \geq 19$
5. $17x_1 + x_2 \geq 13$
6. $7x_4 + 5x_5 \geq 29$
7. $x_2 + 14x_5 \geq 31$
8. $17x_1 + 5x_5 \geq 10$
9. $5x_0 + 5x_5 \geq 13$
10. $5x_0 + 7x_4 \geq 12$
11. $8x_3 + 5x_5 \geq 29$
12. $5x_0 + 17x_1 + x_2 + 8x_3 + 7x_4 + 5x_5 \geq 29$
13. $12x_2 + 14x_5 \geq 16$
14. $12x_2 + 8x_3 \geq 27$
15. $19x_0 + 18x_1 + 12x_2 + 8x_3 + 17x_4 + 14x_5 \geq 27$
16. $2x_1 - 3x_4 \geq 0$
17. $8x_0 - 2x_4 \geq 0$
18. $6x_1 - 3x_5 \geq 0$
19. $17x_1 + x_2 \leq 122$
20. $5x_0 + x_2 \leq 149$
21. $5x_0 + 7x_4 \leq 110$
22. $17x_1 + 8x_3 \leq 36$
23. $8x_3 + 5x_5 \leq 84$
24. $x_2 + 5x_5 \leq 37$
25. $17x_1 + 5x_5 \leq 72$
26. $8x_3 + 7x_4 \leq 38$
27. $18x_1 + 12x_2 + 17x_4 \leq 69$
28. $18x_1 + 17x_4 + 14x_5 \leq 81$
29. $19x_0 + 17x_4 + 14x_5 \leq 83$
30. $19x_0 + 18x_1 + 17x_4 \leq 200$
31. $19x_0 + 18x_1 + 8x_3 \leq 45$
32. $19x_0 + 12x_2 + 17x_4 \leq 59$

## 5: Specify variable bounds and types
- $x_0$ : continuous
- $x_1$ : continuous
- $x_2$ : integer
- $x_3$ : integer
- $x_4$ : continuous
- $x_5$ : integer

## 6: Write the Gurobi code
```python
import gurobi as gp

# Define the model
m = gp.Model()

# Define the variables
x0 = m.addVar(name="x0", lb=0, ub=None, vtype=gp.GRB.CONTINUOUS)  # hours worked by Hank
x1 = m.addVar(name="x1", lb=0, ub=None, vtype=gp.GRB.CONTINUOUS)  # hours worked by Paul
x2 = m.addVar(name="x2", lb=0, ub=None, vtype=gp.GRB.INTEGER)    # hours worked by Mary
x3 = m.addVar(name="x3", lb=0, ub=None, vtype=gp.GRB.INTEGER)    # hours worked by Ringo
x4 = m.addVar(name="x4", lb=0, ub=None, vtype=gp.GRB.CONTINUOUS) # hours worked by Jean
x5 = m.addVar(name="x5", lb=0, ub=None, vtype=gp.GRB.INTEGER)    # hours worked by John

# Define the objective function
m.setObjective(1.73*x0 + 1.02*x1 + 2.71*x2 + 3.81*x3 + 9.75*x4 + 3.15*x5, gp.GRB.MINIMIZE)

# Add constraints
m.addConstr(17*x1 + 8*x3 >= 32)
m.addConstr(5*x0 + 17*x1 >= 22)
m.addConstr(8*x3 + 7*x4 >= 18)
m.addConstr(5*x0 + 8*x3 >= 19)
m.addConstr(17*x1 + x2 >= 13)
m.addConstr(7*x4 + 5*x5 >= 29)
m.addConstr(x2 + 14*x5 >= 31)
m.addConstr(17*x1 + 5*x5 >= 10)
m.addConstr(5*x0 + 5*x5 >= 13)
m.addConstr(5*x0 + 7*x4 >= 12)
m.addConstr(8*x3 + 5*x5 >= 29)
m.addConstr(5*x0 + 17*x1 + x2 + 8*x3 + 7*x4 + 5*x5 >= 29)
m.addConstr(12*x2 + 14*x5 >= 16)
m.addConstr(12*x2 + 8*x3 >= 27)
m.addConstr(19*x0 + 18*x1 + 12*x2 + 8*x3 + 17*x4 + 14*x5 >= 27)
m.addConstr(2*x1 - 3*x4 >= 0)
m.addConstr(8*x0 - 2*x4 >= 0)
m.addConstr(6*x1 - 3*x5 >= 0)
m.addConstr(17*x1 + x2 <= 122)
m.addConstr(5*x0 + x2 <= 149)
m.addConstr(5*x0 + 7*x4 <= 110)
m.addConstr(17*x1 + 8*x3 <= 36)
m.addConstr(8*x3 + 5*x5 <= 84)
m.addConstr(x2 + 5*x5 <= 37)
m.addConstr(17*x1 + 5*x5 <= 72)
m.addConstr(8*x3 + 7*x4 <= 38)
m.addConstr(18*x1 + 12*x2 + 17*x4 <= 69)
m.addConstr(18*x1 + 17*x4 + 14*x5 <= 81)
m.addConstr(19*x0 + 17*x4 + 14*x5 <= 83)
m.addConstr(19*x0 + 18*x1 + 17*x4 <= 200)
m.addConstr(19*x0 + 18*x1 + 8*x3 <= 45)
m.addConstr(19*x0 + 12*x2 + 17*x4 <= 59)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.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)
    print("x5: ", x5.varValue)
else:
    print("No optimal solution found")
```

## 7: Symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Hank'),
        ('x1', 'hours worked by Paul'),
        ('x2', 'hours worked by Mary'),
        ('x3', 'hours worked by Ringo'),
        ('x4', 'hours worked by Jean'),
        ('x5', 'hours worked by John')
    ],
    'objective_function': '1.73*x0 + 1.02*x1 + 2.71*x2 + 3.81*x3 + 9.75*x4 + 3.15*x5',
    'constraints': [
        '17*x1 + 8*x3 >= 32',
        '5*x0 + 17*x1 >= 22',
        '8*x3 + 7*x4 >= 18',
        '5*x0 + 8*x3 >= 19',
        '17*x1 + x2 >= 13',
        '7*x4 + 5*x5 >= 29',
        'x2 + 14*x5 >= 31',
        '17*x1 + 5*x5 >= 10',
        '5*x0 + 5*x5 >= 13',
        '5*x0 + 7*x4 >= 12',
        '8*x3 + 5*x5 >= 29',
        '5*x0 + 17*x1 + x2 + 8*x3 + 7*x4 + 5*x5 >= 29',
        '12*x2 + 14*x5 >= 16',
        '12*x2 + 8*x3 >= 27',
        '19*x0 + 18*x1 + 12*x2 + 8*x3 + 17*x4 + 14*x5 >= 27',
        '2*x1 - 3*x4 >= 0',
        '8*x0 - 2*x4 >= 0',
        '6*x1 - 3*x5 >= 0',
        '17*x1 + x2 <= 122',
        '5*x0 + x2 <= 149',
        '5*x0 + 7*x4 <= 110',
        '17*x1 + 8*x3 <= 36',
        '8*x3 + 5*x5 <= 84',
        'x2 + 5*x5 <= 37',
        '17*x1 + 5*x5 <= 72',
        '8*x3 + 7*x4 <= 38',
        '18*x1 + 12*x2 + 17*x4 <= 69',
        '18*x1 + 17*x4 + 14*x5 <= 81',
        '19*x0 + 17*x4 + 14*x5 <= 83',
        '19*x0 + 18*x1 + 17*x4 <= 200',
        '19*x0 + 18*x1 + 8*x3 <= 45',
        '19*x0 + 12*x2 + 17*x4 <= 59'
    ]
}
```