## Step 1: Define the symbolic representation of the problem
First, let's define the symbolic variables and their corresponding natural language objects:
- $x_0$ = hamburgers
- $x_1$ = black beans
- $x_2$ = hot dogs

## Step 2: Formulate the objective function
The objective function to maximize is $3x_0 + 1x_1 + 5x_2$.

## Step 3: List the constraints
1. $24x_0 \geq 0$ (Implicit, as $x_0 \geq 0$)
2. $21x_0 \geq 0$ (Implicit, as $x_0 \geq 0$)
3. $13x_1 \geq 0$ (Implicit, as $x_1 \geq 0$)
4. $25x_1 \geq 0$ (Implicit, as $x_1 \geq 0$)
5. $16x_2 \geq 0$ (Implicit, as $x_2 \geq 0$)
6. $24x_2 \geq 0$ (Implicit, as $x_2 \geq 0$)
7. $24x_0 + 13x_1 \geq 87$
8. $13x_1 + 16x_2 \geq 30$
9. $21x_0 + 25x_1 \geq 38$
10. $25x_1 + 24x_2 \geq 41$
11. $13x_1 + 16x_2 \leq 190$
12. $24x_0 + 13x_1 \leq 148$
13. $24x_0 + 13x_1 + 16x_2 \leq 148$
14. $21x_0 + 24x_2 \leq 87$
15. $25x_1 + 24x_2 \leq 72$
16. $21x_0 + 25x_1 \leq 120$
17. $21x_0 + 25x_1 + 24x_2 \leq 120$
18. $x_0$ is an integer
19. $x_1$ is an integer
20. $x_2$ can be any real number

## 4: Convert to Gurobi Code
```python
import gurobi

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

# Define the variables
x0 = m.addVar(name="hamburgers", vtype=gurobi.GRB.INTEGER)
x1 = m.addVar(name="black_beans", vtype=gurobi.GRB.INTEGER)
x2 = m.addVar(name="hot_dogs")

# Define the objective function
m.setObjective(3*x0 + x1 + 5*x2, gurobi.GRB.MAXIMIZE)

# Add constraints
m.addConstr(24*x0 + 13*x1 >= 87)
m.addConstr(13*x1 + 16*x2 >= 30)
m.addConstr(21*x0 + 25*x1 >= 38)
m.addConstr(25*x1 + 24*x2 >= 41)
m.addConstr(13*x1 + 16*x2 <= 190)
m.addConstr(24*x0 + 13*x1 <= 148)
m.addConstr(24*x0 + 13*x1 + 16*x2 <= 148)
m.addConstr(21*x0 + 24*x2 <= 87)
m.addConstr(25*x1 + 24*x2 <= 72)
m.addConstr(21*x0 + 25*x1 <= 120)
m.addConstr(21*x0 + 25*x1 + 24*x2 <= 120)

# Optimize the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Hamburgers: ", x0.varValue)
    print("Black Beans: ", x1.varValue)
    print("Hot Dogs: ", x2.varValue)
else:
    print("The model is infeasible")
```

## 5: Symbolic Representation
```json
{
    'sym_variables': [('x0', 'hamburgers'), ('x1', 'black beans'), ('x2', 'hot dogs')],
    'objective_function': '3*x0 + 1*x1 + 5*x2',
    'constraints': [
        '24*x0 + 13*x1 >= 87',
        '13*x1 + 16*x2 >= 30',
        '21*x0 + 25*x1 >= 38',
        '25*x1 + 24*x2 >= 41',
        '13*x1 + 16*x2 <= 190',
        '24*x0 + 13*x1 <= 148',
        '24*x0 + 13*x1 + 16*x2 <= 148',
        '21*x0 + 24*x2 <= 87',
        '25*x1 + 24*x2 <= 72',
        '21*x0 + 25*x1 <= 120',
        '21*x0 + 25*x1 + 24*x2 <= 120',
        'x0 is an integer',
        'x1 is an integer'
    ]
}
```