## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Dale' and 'hours worked by John', which we can denote as $x_1$ and $x_2$ respectively.

## Step 2: Define the objective function in symbolic notation
The objective function to minimize is $8.6x_1x_2 + 8.47x_2^2 + 3.71x_1$.

## Step 3: Define the constraints in symbolic notation
The constraints are:
1. $8x_1^2 + 13x_2^2 \geq 22$
2. $8x_1 + 13x_2 \geq 22$
3. $15x_1 + 13x_2 \geq 39$
4. $2x_1 - x_2 \geq 0$
5. $8x_1^2 + 13x_2^2 \leq 79$
6. $15x_1 + 13x_2 \leq 43$
7. $x_1$ is an integer
8. $x_2$ can be any real number

## 4: Convert the problem into a Gurobi model
We will use Gurobi's Python API to model and solve this problem.

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

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

# Define the variables
x1 = m.addVar(name="hours_worked_by_Dale", vtype=gp.GRB.INTEGER)
x2 = m.addVar(name="hours_worked_by_John")

# Define the objective function
m.setObjective(8.6*x1*x2 + 8.47*x2**2 + 3.71*x1, gp.GRB.MINIMIZE)

# Define the constraints
m.addConstr(8*x1**2 + 13*x2**2 >= 22)
m.addConstr(8*x1 + 13*x2 >= 22)
m.addConstr(15*x1 + 13*x2 >= 39)
m.addConstr(2*x1 - x2 >= 0)
m.addConstr(8*x1**2 + 13*x2**2 <= 79)
m.addConstr(15*x1 + 13*x2 <= 43)

# Optimize the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Hours worked by Dale: {x1.varValue}")
    print(f"Hours worked by John: {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 Dale'), ('x2', 'hours worked by John')],
    'objective_function': '8.6*x1*x2 + 8.47*x2^2 + 3.71*x1',
    'constraints': [
        '8*x1^2 + 13*x2^2 >= 22',
        '8*x1 + 13*x2 >= 22',
        '15*x1 + 13*x2 >= 39',
        '2*x1 - x2 >= 0',
        '8*x1^2 + 13*x2^2 <= 79',
        '15*x1 + 13*x2 <= 43',
        'x1 is an integer'
    ]
}
```