## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Bill', 'hours worked by Bobby', and 'hours worked by Paul'. Let's denote them as $x_0$, $x_1$, and $x_2$ respectively.

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

## Step 3: Define the constraints in symbolic notation
The constraints are:
- $2x_0 \leq 74$
- $12x_0 \leq 145$
- $11x_1 \leq 74$
- $10x_1 \leq 145$
- $12x_2 \leq 74$
- $10x_2 \leq 145$
- $2x_0 + 12x_2 \geq 11$
- $11x_1 + 12x_2 \geq 11$
- $2x_0 + 11x_1 + 12x_2 \geq 11$
- $10x_1 + 10x_2 \geq 24$
- $12x_0 + 10x_1 \geq 37$
- $12x_0 + 10x_2 \geq 22$
- $12x_0 + 10x_1 + 10x_2 \geq 37$
- $12x_0 + 10x_1 + 10x_2 \leq 141$
- $x_0 - 10x_1 \geq 0$
- $2x_0 + 11x_1 \leq 51$
- $2x_0 + 11x_1 + 12x_2 \leq 29$
- $12x_0 + 10x_2 \leq 53$

## Step 4: Convert the problem into a Gurobi code
```python
import gurobi

# Create a new model
m = gurobi.Model()

# Define the variables
x0 = m.addVar(name="hours_worked_by_Bill", lb=0)
x1 = m.addVar(name="hours_worked_by_Bobby", lb=0)
x2 = m.addVar(name="hours_worked_by_Paul", lb=0)

# Set the objective function
m.setObjective(3*x0 + 3*x1 + x2, gurobi.GRB.MINIMIZE)

# Add constraints
m.addConstr(2*x0 <= 74)
m.addConstr(12*x0 <= 145)
m.addConstr(11*x1 <= 74)
m.addConstr(10*x1 <= 145)
m.addConstr(12*x2 <= 74)
m.addConstr(10*x2 <= 145)
m.addConstr(2*x0 + 12*x2 >= 11)
m.addConstr(11*x1 + 12*x2 >= 11)
m.addConstr(2*x0 + 11*x1 + 12*x2 >= 11)
m.addConstr(10*x1 + 10*x2 >= 24)
m.addConstr(12*x0 + 10*x1 >= 37)
m.addConstr(12*x0 + 10*x2 >= 22)
m.addConstr(12*x0 + 10*x1 + 10*x2 >= 37)
m.addConstr(12*x0 + 10*x1 + 10*x2 <= 141)
m.addConstr(x0 - 10*x1 >= 0)
m.addConstr(2*x0 + 11*x1 <= 51)
m.addConstr(2*x0 + 11*x1 + 12*x2 <= 29)
m.addConstr(12*x0 + 10*x2 <= 53)

# Optimize the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Hours worked by Bill: ", x0.varValue)
    print("Hours worked by Bobby: ", x1.varValue)
    print("Hours worked by Paul: ", x2.varValue)
else:
    print("The model is infeasible")
```

## Step 5: Provide the symbolic representation of the problem
```json
{
    'sym_variables': [('x0', 'hours worked by Bill'), ('x1', 'hours worked by Bobby'), ('x2', 'hours worked by Paul')],
    'objective_function': '3*x0 + 3*x1 + x2',
    'constraints': [
        '2*x0 <= 74',
        '12*x0 <= 145',
        '11*x1 <= 74',
        '10*x1 <= 145',
        '12*x2 <= 74',
        '10*x2 <= 145',
        '2*x0 + 12*x2 >= 11',
        '11*x1 + 12*x2 >= 11',
        '2*x0 + 11*x1 + 12*x2 >= 11',
        '10*x1 + 10*x2 >= 24',
        '12*x0 + 10*x1 >= 37',
        '12*x0 + 10*x2 >= 22',
        '12*x0 + 10*x1 + 10*x2 >= 37',
        '12*x0 + 10*x1 + 10*x2 <= 141',
        'x0 - 10*x1 >= 0',
        '2*x0 + 11*x1 <= 51',
        '2*x0 + 11*x1 + 12*x2 <= 29',
        '12*x0 + 10*x2 <= 53'
    ]
}
```