## Step 1: Define the symbolic representation of the problem
Let's denote the amount of Fruity Loop detergent to be produced as $x_1$ and the amount of Passion Cook detergent as $x_2$. The objective is to minimize the cost, where Fruity Loop costs $6 per kg and Passion Cook costs $5 per kg. Therefore, the objective function is $6x_1 + 5x_2$. The constraints are that the total amount of soap used must be at least 20 kg and the total amount of citric acid used must be at least 15 kg. Given that Fruity Loop consists of 10% soap and 6% citric acid, and Passion Cook consists of 5% soap and 10% citric acid, we have:
- $0.10x_1 + 0.05x_2 \geq 20$ (soap constraint)
- $0.06x_1 + 0.10x_2 \geq 15$ (citric acid constraint)
Also, $x_1 \geq 0$ and $x_2 \geq 0$ because the amounts of detergents cannot be negative.

## 2: Convert the problem into a symbolic representation
The symbolic variables are:
- $x_1$ for Fruity Loop
- $x_2$ for Passion Cook

The objective function is: $6x_1 + 5x_2$

The constraints are:
- $0.10x_1 + 0.05x_2 \geq 20$
- $0.06x_1 + 0.10x_2 \geq 15$
- $x_1 \geq 0$
- $x_2 \geq 0$

## 3: Represent the problem in the requested JSON format
```json
{
'sym_variables': [('x1', 'Fruity Loop'), ('x2', 'Passion Cook')],
'objective_function': '6*x1 + 5*x2',
'constraints': [
    '0.10*x1 + 0.05*x2 >= 20',
    '0.06*x1 + 0.10*x2 >= 15',
    'x1 >= 0',
    'x2 >= 0'
]
}
```

## 4: Implement the problem in Gurobi code
```python
import gurobi

def solve_problem():
    # Create a new model
    model = gurobi.Model()

    # Define the variables
    x1 = model.addVar(name="Fruity_Loop", lb=0, ub=gurobi.GRB.INFINITY)
    x2 = model.addVar(name="Passion_Cook", lb=0, ub=gurobi.GRB.INFINITY)

    # Define the objective function
    model.setObjective(6 * x1 + 5 * x2, gurobi.GRB.MINIMIZE)

    # Define the constraints
    model.addConstr(0.10 * x1 + 0.05 * x2 >= 20, name="soap_constraint")
    model.addConstr(0.06 * x1 + 0.10 * x2 >= 15, name="citric_acid_constraint")

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Fruity Loop: {x1.varValue} kg")
        print(f"Passion Cook: {x2.varValue} kg")
        print(f"Minimum cost: ${model.objVal:.2f}")
    else:
        print("No optimal solution found.")

solve_problem()
```