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

## Step 2: Define the objective function in symbolic notation
The objective function to maximize is $7.41x_1 + 3.98x_2$.

## Step 3: Define the constraints in symbolic notation
The constraints are:
1. $7x_1 + 6x_2 \geq 34$
2. $6x_1 + 5x_2 \geq 31$
3. $-9x_1 + 3x_2 \geq 0$
4. $7x_1 + 6x_2 \leq 75$
5. $6x_1 + 5x_2 \leq 38$

## 4: Convert the problem into a Gurobi model
We will use Gurobi to solve this linear programming problem.

## Step 5: Write the Gurobi code
```python
import gurobi as gp

# Create a new model
m = gp.Model("optimization_problem")

# Define the variables
x1 = m.addVar(name="hours_worked_by_Paul", lb=0)  # hours worked by Paul
x2 = m.addVar(name="hours_worked_by_Peggy", lb=0)  # hours worked by Peggy

# Define the objective function
m.setObjective(7.41 * x1 + 3.98 * x2, gp.GRB.MAXIMIZE)

# Define the constraints
m.addConstr(7 * x1 + 6 * x2 >= 34, name="min_dollar_cost")
m.addConstr(6 * x1 + 5 * x2 >= 31, name="min_productivity")
m.addConstr(-9 * x1 + 3 * x2 >= 0, name="peggy_vs_paul")
m.addConstr(7 * x1 + 6 * x2 <= 75, name="max_dollar_cost")
m.addConstr(6 * x1 + 5 * x2 <= 38, name="max_productivity")

# Optimize the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Hours worked by Paul: {x1.varValue}")
    print(f"Hours worked by Peggy: {x2.varValue}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found.")
```

## 6: Provide the symbolic representation of the problem
```json
{
    'sym_variables': [('x1', 'hours worked by Paul'), ('x2', 'hours worked by Peggy')],
    'objective_function': '7.41*x1 + 3.98*x2',
    'constraints': [
        '7*x1 + 6*x2 >= 34',
        '6*x1 + 5*x2 >= 31',
        '-9*x1 + 3*x2 >= 0',
        '7*x1 + 6*x2 <= 75',
        '6*x1 + 5*x2 <= 38'
    ]
}
```