## Problem Description and Formulation

The problem is a classic example of a linear programming problem. The family business wants to maximize their profit by producing the optimal number of apple pies and blueberry pies given certain constraints.

Let's define the decision variables:

* $x$: the number of apple pies produced
* $y$: the number of blueberry pies produced

The objective function is to maximize the total profit:

* Profit per apple pie: $5
* Profit per blueberry pie: $6
* Total profit: $5x + 6y

The constraints are:

* Time constraint: The family business operates for 4500 minutes per week.
	+ Time to make one apple pie: 30 minutes
	+ Time to make one blueberry pie: 40 minutes
	+ Total time: $30x + 40y \leq 4500$
* Fruit availability constraint:
	+ Apple pies: $x \leq 70$
	+ Blueberry pies: $y \leq 60$
* Non-negativity constraint:
	+ $x \geq 0$
	+ $y \geq 0$

## Gurobi Code

```python
import gurobi

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

# Define the decision variables
x = m.addVar(lb=0, ub=70, name="apple_pies")
y = m.addVar(lb=0, ub=60, name="blueberry_pies")

# Define the objective function
m.setObjective(5 * x + 6 * y, gurobi.GRB.MAXIMIZE)

# Define the constraints
m.addConstr(30 * x + 40 * y <= 4500, name="time_constraint")

# Solve the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Apple pies: {x.varValue}")
    print(f"Blueberry pies: {y.varValue}")
    print(f"Max profit: {m.objVal}")
else:
    print("No optimal solution found.")
```

This code defines the decision variables, objective function, and constraints, and then solves the model using Gurobi. The solution is then printed out, including the optimal number of apple pies and blueberry pies to produce, and the maximum profit.