## Step 1: Define the symbolic representation of the variables
The variables are ['hours worked by Peggy', 'hours worked by Bobby', 'hours worked by Ringo', 'hours worked by Laura']. Let's denote them as $x_0$, $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Define the symbolic representation of the problem
The objective function to maximize is $7x_0 + 8x_1 + 1x_2 + 1x_3$.

## Step 3: List the constraints
The constraints are:
- $1x_0 + 13x_2 + 11x_3 \geq 24$
- $1x_0 + 4x_1 + 11x_3 \geq 24$
- $1x_0 + 4x_1 + 13x_2 \geq 24$
- $1x_0 + 13x_2 + 11x_3 \geq 18$
- $1x_0 + 4x_1 + 11x_3 \geq 18$
- $1x_0 + 4x_1 + 13x_2 \geq 18$
- $1x_0 + 13x_2 + 11x_3 \geq 32$
- $1x_0 + 4x_1 + 11x_3 \geq 32$
- $1x_0 + 4x_1 + 13x_2 \geq 32$
- $12x_0 + 11x_2 \geq 30$
- $10x_1 + 10x_3 \geq 23$
- $12x_0 + 10x_1 + 10x_3 \geq 20$
- $1x_0 + 4x_1 \leq 110$
- $13x_2 + 11x_3 \leq 54$
- $1x_0 + 4x_1 + 13x_2 + 11x_3 \leq 54$
- $12x_0 + 10x_3 \leq 98$
- $12x_0 + 11x_2 \leq 50$
- $10x_1 + 10x_3 \leq 78$
- $11x_2 + 10x_3 \leq 110$
- $12x_0 + 10x_1 \leq 32$
- $12x_0 + 10x_1 + 10x_3 \leq 109$
- $12x_0 + 10x_1 + 11x_2 + 10x_3 \leq 109$

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

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

# Define the variables
x0 = m.addVar(name='hours_worked_by_Peggy', lb=0)
x1 = m.addVar(name='hours_worked_by_Bobby', lb=0)
x2 = m.addVar(name='hours_worked_by_Ringo', lb=0)
x3 = m.addVar(name='hours_worked_by_Laura', lb=0)

# Define the objective function
m.setObjective(7 * x0 + 8 * x1 + x2 + x3, gurobi.GRB.MAXIMIZE)

# Add constraints
m.addConstr(1 * x0 + 13 * x2 + 11 * x3 >= 24)
m.addConstr(1 * x0 + 4 * x1 + 11 * x3 >= 24)
m.addConstr(1 * x0 + 4 * x1 + 13 * x2 >= 24)
m.addConstr(1 * x0 + 13 * x2 + 11 * x3 >= 18)
m.addConstr(1 * x0 + 4 * x1 + 11 * x3 >= 18)
m.addConstr(1 * x0 + 4 * x1 + 13 * x2 >= 18)
m.addConstr(1 * x0 + 13 * x2 + 11 * x3 >= 32)
m.addConstr(1 * x0 + 4 * x1 + 11 * x3 >= 32)
m.addConstr(1 * x0 + 4 * x1 + 13 * x2 >= 32)
m.addConstr(12 * x0 + 11 * x2 >= 30)
m.addConstr(10 * x1 + 10 * x3 >= 23)
m.addConstr(12 * x0 + 10 * x1 + 10 * x3 >= 20)
m.addConstr(1 * x0 + 4 * x1 <= 110)
m.addConstr(13 * x2 + 11 * x3 <= 54)
m.addConstr(1 * x0 + 4 * x1 + 13 * x2 + 11 * x3 <= 54)
m.addConstr(12 * x0 + 10 * x3 <= 98)
m.addConstr(12 * x0 + 11 * x2 <= 50)
m.addConstr(10 * x1 + 10 * x3 <= 78)
m.addConstr(11 * x2 + 10 * x3 <= 110)
m.addConstr(12 * x0 + 10 * x1 <= 32)
m.addConstr(12 * x0 + 10 * x1 + 10 * x3 <= 109)
m.addConstr(12 * x0 + 10 * x1 + 11 * x2 + 10 * x3 <= 109)

# Optimize the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print('Objective: ', m.objVal)
    print('Hours worked by Peggy: ', x0.varValue)
    print('Hours worked by Bobby: ', x1.varValue)
    print('Hours worked by Ringo: ', x2.varValue)
    print('Hours worked by Laura: ', x3.varValue)
else:
    print('The problem is infeasible')
```

## 5: Symbolic representation of the problem
```json
{
    'sym_variables': [
        ['x0', 'hours worked by Peggy'],
        ['x1', 'hours worked by Bobby'],
        ['x2', 'hours worked by Ringo'],
        ['x3', 'hours worked by Laura']
    ],
    'objective_function': '7*x0 + 8*x1 + x2 + x3',
    'constraints': [
        '1*x0 + 13*x2 + 11*x3 >= 24',
        '1*x0 + 4*x1 + 11*x3 >= 24',
        '1*x0 + 4*x1 + 13*x2 >= 24',
        '1*x0 + 13*x2 + 11*x3 >= 18',
        '1*x0 + 4*x1 + 11*x3 >= 18',
        '1*x0 + 4*x1 + 13*x2 >= 18',
        '1*x0 + 13*x2 + 11*x3 >= 32',
        '1*x0 + 4*x1 + 11*x3 >= 32',
        '1*x0 + 4*x1 + 13*x2 >= 32',
        '12*x0 + 11*x2 >= 30',
        '10*x1 + 10*x3 >= 23',
        '12*x0 + 10*x1 + 10*x3 >= 20',
        '1*x0 + 4*x1 <= 110',
        '13*x2 + 11*x3 <= 54',
        '1*x0 + 4*x1 + 13*x2 + 11*x3 <= 54',
        '12*x0 + 10*x3 <= 98',
        '12*x0 + 11*x2 <= 50',
        '10*x1 + 10*x3 <= 78',
        '11*x2 + 10*x3 <= 110',
        '12*x0 + 10*x1 <= 32',
        '12*x0 + 10*x1 + 10*x3 <= 109',
        '12*x0 + 10*x1 + 11*x2 + 10*x3 <= 109'
    ]
}
```