## Problem Description and Formulation

The problem is an optimization problem where we need to maximize the objective function:

\[ 3.78x_0 + 3.66x_1 + 3.88x_2 \]

subject to several constraints. The variables \(x_0\), \(x_1\), and \(x_2\) represent the milligrams of vitamin C, vitamin A, and vitamin B7, respectively.

## Constraints

1. The kidney support index of \(x_0\) is 2.
2. The muscle growth index of \(x_0\) is 1.
3. The kidney support index of \(x_1\) is 3.
4. The muscle growth index of \(x_1\) is 4.
5. The kidney support index of \(x_2\) is 3.
6. The muscle growth index of \(x_2\) is 7.
7. \(2x_0 + 3x_1 \geq 12\)
8. \(4x_1 + 7x_2 \geq 35\)
9. \(3x_1 + 3x_2 \leq 48\)
10. \(2x_0 + 3x_1 \leq 35\)
11. \(2x_0 + 3x_1 + 3x_2 \leq 35\)
12. \(x_0 + 4x_1 \leq 112\)
13. \(x_0 + 7x_2 \leq 79\)
14. \(x_0 + 4x_1 + 7x_2 \leq 79\)
15. \(x_0\) is continuous.
16. \(x_1\) is an integer.
17. \(x_2\) is continuous.

## Gurobi Code

```python
import gurobi

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

    # Define variables
    x0 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="vitamin_C")
    x1 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="vitamin_A", vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="vitamin_B7")

    # Objective function
    model.setObjective(3.78 * x0 + 3.66 * x1 + 3.88 * x2, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(2 * x0 + 3 * x1 >= 12, name="kidney_support_index_constraint1")
    model.addConstr(4 * x1 + 7 * x2 >= 35, name="muscle_growth_index_constraint1")
    model.addConstr(3 * x1 + 3 * x2 <= 48, name="kidney_support_index_constraint2")
    model.addConstr(2 * x0 + 3 * x1 <= 35, name="kidney_support_index_constraint3")
    model.addConstr(2 * x0 + 3 * x1 + 3 * x2 <= 35, name="kidney_support_index_constraint4")
    model.addConstr(x0 + 4 * x1 <= 112, name="muscle_growth_index_constraint2")
    model.addConstr(x0 + 7 * x2 <= 79, name="muscle_growth_index_constraint3")
    model.addConstr(x0 + 4 * x1 + 7 * x2 <= 79, name="muscle_growth_index_constraint4")

    # Update model
    model.update()

    # Optimize model
    model.optimize()

    # Print solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Milligrams of vitamin C: {x0.varValue}")
        print(f"Milligrams of vitamin A: {x1.varValue}")
        print(f"Milligrams of vitamin B7: {x2.varValue}")
        print(f"Objective function value: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```