## Step 1: Define the optimization problem
The problem is to minimize the objective function: $7 \times \text{black beans} + 8 \times \text{bananas} + 4 \times \text{corn cobs} + 7 \times \text{hamburgers}$.

## Step 2: List all the constraints
The constraints are:
- Healthiness rating constraints:
  - $5.97 \times \text{bananas} + 9.6 \times \text{corn cobs} \geq 24$
  - $10.38 \times \text{black beans} + 9.6 \times \text{corn cobs} \geq 16$
  - $10.38 \times \text{black beans} + 5.97 \times \text{bananas} + 1.28 \times \text{hamburgers} \geq 25$
  - $10.38 \times \text{black beans} + 5.97 \times \text{bananas} + 9.6 \times \text{corn cobs} + 1.28 \times \text{hamburgers} \geq 25$

- Carbohydrates constraints:
  - $8.04 \times \text{black beans} + 3.8 \times \text{bananas} \geq 23$
  - $8.04 \times \text{black beans} + 9.58 \times \text{hamburgers} \geq 39$
  - $8.04 \times \text{black beans} + 7.77 \times \text{corn cobs} \geq 23$
  - $8.04 \times \text{black beans} + 3.8 \times \text{bananas} + 9.58 \times \text{hamburgers} \geq 28$
  - $8.04 \times \text{black beans} + 7.77 \times \text{corn cobs} + 9.58 \times \text{hamburgers} \geq 28$
  - $8.04 \times \text{black beans} + 3.8 \times \text{bananas} + 9.58 \times \text{hamburgers} \geq 32$
  - $8.04 \times \text{black beans} + 7.77 \times \text{corn cobs} + 9.58 \times \text{hamburgers} \geq 32$
  - $8.04 \times \text{black beans} + 3.8 \times \text{bananas} + 7.77 \times \text{corn cobs} + 9.58 \times \text{hamburgers} \geq 32$
  - $3.8 \times \text{bananas} + 7.77 \times \text{corn cobs} \leq 55$
  - $8.04 \times \text{black beans} + 9.58 \times \text{hamburgers} \leq 84$
  - $7.77 \times \text{corn cobs} + 9.58 \times \text{hamburgers} \leq 123$
  - $3.8 \times \text{bananas} + 9.58 \times \text{hamburgers} \leq 154$
  - $3.8 \times \text{bananas} + 7.77 \times \text{corn cobs} + 9.58 \times \text{hamburgers} \leq 172$
  - $8.04 \times \text{black beans} + 3.8 \times \text{bananas} + 9.58 \times \text{hamburgers} \leq 69$
  - $8.04 \times \text{black beans} + 7.77 \times \text{corn cobs} + 9.58 \times \text{hamburgers} \leq 104$

- Calcium constraints:
  - $7.25 \times \text{bananas} + 1.68 \times \text{corn cobs} \geq 29$
  - $2.96 \times \text{black beans} + 9.9 \times \text{hamburgers} \geq 30$
  - $2.96 \times \text{black beans} + 1.68 \times \text{corn cobs} \geq 19$
  - $2.96 \times \text{black beans} + 7.25 \times \text{bananas} + 1.68 \times \text{corn cobs} \geq 30$
  - $2.96 \times \text{black beans} + 7.25 \times \text{bananas} + 1.68 \times \text{corn cobs} + 9.9 \times \text{hamburgers} \geq 30$
  - $2.96 \times \text{black beans} + 9.9 \times \text{hamburgers} \leq 48$
  - $2.96 \times \text{black beans} + 7.25 \times \text{bananas} \leq 163$
  - $7.25 \times \text{bananas} + 1.68 \times \text{corn cobs} \leq 88$
  - $2.96 \times \text{black beans} + 1.68 \times \text{corn cobs} \leq 80$
  - $1.68 \times \text{corn cobs} + 9.9 \times \text{hamburgers} \leq 137$
  - $7.25 \times \text{bananas} + 9.9 \times \text{hamburgers} \leq 62$

- Tastiness rating constraints:
  - $4.45 \times \text{bananas} + 5.45 \times \text{corn cobs} \geq 23$
  - $8.47 \times \text{black beans} + 5.45 \times \text{corn cobs} \geq 20$
  - $8.47 \times \text{black beans} + 0.76 \times \text{hamburgers} \geq 24$
  - $5.45 \times \text{corn cobs} + 0.76 \times \text{hamburgers} \geq 36$
  - $8.47 \times \text{black beans} + 4.45 \times \text{bananas} + 0.76 \times \text{hamburgers} \geq 28$
  - $8.47 \times \text{black beans} + 4.45 \times \text{bananas} + 5.45 \times \text{corn cobs} + 0.76 \times \text{hamburgers} \geq 28$
  - $8.47 \times \text{black beans} + 5.45 \times \text{corn cobs} + 0.76 \times \text{hamburgers} \leq 151$

- Fiber constraints:
  - $8.75 \times \text{black beans} + 5.16 \times \text{bananas} \geq 29$
  - $8.75 \times \text{black beans} + 5.16 \times \text{bananas} + 2.4 \times \text{corn cobs} + 8.76 \times \text{hamburgers} \geq 29$
  - $8.75 \times \text{black beans} + 8.76 \times \text{hamburgers} \leq 213$
  - $5.16 \times \text{bananas} + 8.76 \times \text{hamburgers} \leq 101$
  - $5.16 \times \text{bananas} + 2.4 \times \text{corn cobs} \leq 180$
  - $8.75 \times \text{black beans} + 5.16 \times \text{bananas} + 8.76 \times \text{hamburgers} \leq 75$

- Other constraints:
  - $1 \times \text{bananas} - 2 \times \text{hamburgers} \geq 0$
  - $3 \times \text{black beans} - 10 \times \text{corn cobs} \geq 0$
  - $3 \times \text{corn cobs} - 10 \times \text{hamburgers} \geq 0$

## Step 3: Implement the optimization problem using Gurobi
We will use the Gurobi Python library to model and solve this optimization problem.

```python
import gurobi as gp

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

# Define the variables
black_beans = m.addVar(name="black_beans", lb=0)  # No lower bound specified, assuming 0
bananas = m.addVar(name="bananas", lb=0)  # No lower bound specified, assuming 0
corn_cobs = m.addVar(name="corn_cobs", lb=0)  # No lower bound specified, assuming 0
hamburgers = m.addVar(name="hamburgers", lb=0)  # No lower bound specified, assuming 0

# Objective function
m.setObjective(7 * black_beans + 8 * bananas + 4 * corn_cobs + 7 * hamburgers, gp.GRB.MINIMIZE)

# Healthiness rating constraints
m.addConstr(5.97 * bananas + 9.6 * corn_cobs >= 24)
m.addConstr(10.38 * black_beans + 9.6 * corn_cobs >= 16)
m.addConstr(10.38 * black_beans + 5.97 * bananas + 1.28 * hamburgers >= 25)
m.addConstr(10.38 * black_beans + 5.97 * bananas + 9.6 * corn_cobs + 1.28 * hamburgers >= 25)

# Carbohydrates constraints
m.addConstr(8.04 * black_beans + 3.8 * bananas >= 23)
m.addConstr(8.04 * black_beans + 9.58 * hamburgers >= 39)
m.addConstr(8.04 * black_beans + 7.77 * corn_cobs >= 23)
m.addConstr(8.04 * black_beans + 3.8 * bananas + 9.58 * hamburgers >= 28)
m.addConstr(8.04 * black_beans + 7.77 * corn_cobs + 9.58 * hamburgers >= 28)
m.addConstr(8.04 * black_beans + 3.8 * bananas + 9.58 * hamburgers >= 32)
m.addConstr(8.04 * black_beans + 7.77 * corn_cobs + 9.58 * hamburgers >= 32)
m.addConstr(8.04 * black_beans + 3.8 * bananas + 7.77 * corn_cobs + 9.58 * hamburgers >= 32)
m.addConstr(3.8 * bananas + 7.77 * corn_cobs <= 55)
m.addConstr(8.04 * black_beans + 9.58 * hamburgers <= 84)
m.addConstr(7.77 * corn_cobs + 9.58 * hamburgers <= 123)
m.addConstr(3.8 * bananas + 9.58 * hamburgers <= 154)
m.addConstr(3.8 * bananas + 7.77 * corn_cobs + 9.58 * hamburgers <= 172)
m.addConstr(8.04 * black_beans + 3.8 * bananas + 9.58 * hamburgers <= 69)
m.addConstr(8.04 * black_beans + 7.77 * corn_cobs + 9.58 * hamburgers <= 104)

# Calcium constraints
m.addConstr(7.25 * bananas + 1.68 * corn_cobs >= 29)
m.addConstr(2.96 * black_beans + 9.9 * hamburgers >= 30)
m.addConstr(2.96 * black_beans + 1.68 * corn_cobs >= 19)
m.addConstr(2.96 * black_beans + 7.25 * bananas + 1.68 * corn_cobs >= 30)
m.addConstr(2.96 * black_beans + 7.25 * bananas + 1.68 * corn_cobs + 9.9 * hamburgers >= 30)
m.addConstr(2.96 * black_beans + 9.9 * hamburgers <= 48)
m.addConstr(2.96 * black_beans + 7.25 * bananas <= 163)
m.addConstr(7.25 * bananas + 1.68 * corn_cobs <= 88)
m.addConstr(2.96 * black_beans + 1.68 * corn_cobs <= 80)
m.addConstr(1.68 * corn_cobs + 9.9 * hamburgers <= 137)
m.addConstr(7.25 * bananas + 9.9 * hamburgers <= 62)

# Tastiness rating constraints
m.addConstr(4.45 * bananas + 5.45 * corn_cobs >= 23)
m.addConstr(8.47 * black_beans + 5.45 * corn_cobs >= 20)
m.addConstr(8.47 * black_beans + 0.76 * hamburgers >= 24)
m.addConstr(5.45 * corn_cobs + 0.76 * hamburgers >= 36)
m.addConstr(8.47 * black_beans + 4.45 * bananas + 0.76 * hamburgers >= 28)
m.addConstr(8.47 * black_beans + 4.45 * bananas + 5.45 * corn_cobs + 0.76 * hamburgers >= 28)
m.addConstr(8.47 * black_beans + 5.45 * corn_cobs + 0.76 * hamburgers <= 151)

# Fiber constraints
m.addConstr(8.75 * black_beans + 5.16 * bananas >= 29)
m.addConstr(8.75 * black_beans + 5.16 * bananas + 2.4 * corn_cobs + 8.76 * hamburgers >= 29)
m.addConstr(8.75 * black_beans + 8.76 * hamburgers <= 213)
m.addConstr(5.16 * bananas + 8.76 * hamburgers <= 101)
m.addConstr(5.16 * bananas + 2.4 * corn_cobs <= 180)
m.addConstr(8.75 * black_beans + 5.16 * bananas + 8.76 * hamburgers <= 75)

# Other constraints
m.addConstr(bananas - 2 * hamburgers >= 0)
m.addConstr(3 * black_beans - 10 * corn_cobs >= 0)
m.addConstr(3 * corn_cobs - 10 * hamburgers >= 0)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objval)
    print("Black beans: ", black_beans.varValue)
    print("Bananas: ", bananas.varValue)
    print("Corn cobs: ", corn_cobs.varValue)
    print("Hamburgers: ", hamburgers.varValue)
else:
    print("The model is infeasible.")
```