## Problem Description and Formulation

The problem is a classic example of a linear programming problem. The goal is to maximize the profit of a tea shop by determining the optimal number of cups of green tea and black tea to produce, given certain constraints.

Let's define the decision variables:

* `G`: number of cups of green tea to produce
* `B`: number of cups of black tea to produce

The objective function is to maximize the profit:

* Profit = $2G + $3B

The constraints are:

* Time constraint: 3 minutes to make a cup of green tea and 5 minutes to make a cup of black tea, with a total of 560 minutes available: `3G + 5B <= 560`
* Production constraint: the shop owner only has enough product to make 150 total cups per week: `G + B <= 150`
* Non-negativity constraint: the number of cups of green tea and black tea cannot be negative: `G >= 0, B >= 0`

## Gurobi Code

```python
import gurobi

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

# Define the decision variables
G = m.addVar(lb=0, name="Green Tea")
B = m.addVar(lb=0, name="Black Tea")

# Define the objective function
m.setObjective(2*G + 3*B, gurobi.GRB.MAXIMIZE)

# Define the constraints
m.addConstr(3*G + 5*B <= 560, name="Time Constraint")
m.addConstr(G + B <= 150, name="Production Constraint")

# Optimize the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Green Tea: {G.varValue}")
    print(f"Black Tea: {B.varValue}")
    print(f"Profit: ${2*G.varValue + 3*B.varValue:.2f}")
else:
    print("No optimal solution found.")
```