## Step 1: Define the symbolic representation of the variables
The variables are ['hours worked by Peggy', 'hours worked by Bobby', 'hours worked by John'] which can be represented symbolically as $x_0$, $x_1$, and $x_2$ respectively.

## Step 2: Define the symbolic representation of the problem
The objective function to minimize is $1x_0 + 9x_1 + 4x_2$.

## Step 3: List all the constraints
The constraints given are:
- $7.65x_0 \geq 0$ (implicit, as $x_0$ represents hours and is non-negative)
- $24.5x_0 \geq 0$ (implicit, as $x_0$ represents hours and is non-negative)
- $21.97x_1 \geq 0$ (implicit, as $x_1$ represents hours and is non-negative)
- $3.82x_1 \geq 0$ (implicit, as $x_1$ represents hours and is non-negative)
- $25.24x_2 \geq 0$ (implicit, as $x_2$ represents hours and is non-negative)
- $20.52x_2 \geq 0$ (implicit, as $x_2$ represents hours and is non-negative)
- $7.65x_0 + 21.97x_1 \geq 63$
- $7.65x_0 + 25.24x_2 \geq 54$
- $7.65x_0 + 21.97x_1 + 25.24x_2 \geq 54$
- $3.82x_1 + 20.52x_2 \geq 19$
- $24.5x_0 + 20.52x_2 \geq 22$
- $24.5x_0 + 3.82x_1 + 20.52x_2 \geq 22$
- $-x_0 + 9x_2 \geq 0$
- $x_1 - 2x_2 \geq 0$
- $24.5x_0 + 3.82x_1 \leq 93$
- $24.5x_0 + 20.52x_2 \leq 46$
- $3.82x_1 + 20.52x_2 \leq 108$

## 4: Determine the symbolic representation
The symbolic representation of the problem is:
```json
{
'sym_variables': [
    ('x0', 'hours worked by Peggy'),
    ('x1', 'hours worked by Bobby'),
    ('x2', 'hours worked by John')
],
'objective_function': '1*x0 + 9*x1 + 4*x2',
'constraints': [
    '7.65*x0 + 21.97*x1 >= 63',
    '7.65*x0 + 25.24*x2 >= 54',
    '7.65*x0 + 21.97*x1 + 25.24*x2 >= 54',
    '3.82*x1 + 20.52*x2 >= 19',
    '24.5*x0 + 20.52*x2 >= 22',
    '24.5*x0 + 3.82*x1 + 20.52*x2 >= 22',
    '-1*x0 + 9*x2 >= 0',
    '1*x1 - 2*x2 >= 0',
    '24.5*x0 + 3.82*x1 <= 93',
    '24.5*x0 + 20.52*x2 <= 46',
    '3.82*x1 + 20.52*x2 <= 108'
]
}
```

## 5: Create the Gurobi model
We will now create a Gurobi model to solve this optimization problem.

```python
import gurobi

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

# Define the variables
x0 = m.addVar(name='x0', lb=0, type=gurobi.GRB.INTEGER)  # hours worked by Peggy
x1 = m.addVar(name='x1', lb=0, type=gurobi.GRB.INTEGER)  # hours worked by Bobby
x2 = m.addVar(name='x2', lb=0)  # hours worked by John

# Set the objective function
m.setObjective(1*x0 + 9*x1 + 4*x2, gurobi.GRB.MINIMIZE)

# Add constraints
m.addConstr(7.65*x0 + 21.97*x1 >= 63)
m.addConstr(7.65*x0 + 25.24*x2 >= 54)
m.addConstr(7.65*x0 + 21.97*x1 + 25.24*x2 >= 54)
m.addConstr(3.82*x1 + 20.52*x2 >= 19)
m.addConstr(24.5*x0 + 20.52*x2 >= 22)
m.addConstr(24.5*x0 + 3.82*x1 + 20.52*x2 >= 22)
m.addConstr(-1*x0 + 9*x2 >= 0)
m.addConstr(x1 - 2*x2 >= 0)
m.addConstr(24.5*x0 + 3.82*x1 <= 93)
m.addConstr(24.5*x0 + 20.52*x2 <= 46)
m.addConstr(3.82*x1 + 20.52*x2 <= 108)

# Solve 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 John: ', x2.varValue)
else:
    print('The model is infeasible')
```