## Problem Description and Formulation

The problem is a classic example of a linear programming problem. We have two products: bookcases and garden chairs. Each product requires certain resources: woodworking hours, boxes of nails, and units of varnish. The goal is to maximize profit given the available resources.

Let's define the decision variables:
- \(B\): the number of bookcases to be made
- \(G\): the number of garden chairs to be made

The objective function to maximize profit (\(P\)) is:
\[ P = 270B + 350G \]

The constraints based on the available resources are:
1. Woodworking: \(3B + 2G \leq 80\)
2. Nails: \(2B + 4G \leq 70\)
3. Varnish: \(3B + 3G \leq 90\)
4. Non-negativity: \(B \geq 0, G \geq 0\)

## Gurobi Code

To solve this problem using Gurobi in Python, we will first install the gurobi package if it's not already installed. You can install it via pip:

```bash
pip install gurobi
```

Now, let's formulate and solve the problem using Gurobi:

```python
import gurobi as gp

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

# Define the decision variables
B = m.addVar(name="bookcases", lb=0, vtype=gp.GRB.CONTINUOUS)
G = m.addVar(name="garden_chairs", lb=0, vtype=gp.GRB.CONTINUOUS)

# Objective function: maximize profit
m.setObjective(270*B + 350*G, gp.GRB.MAXIMIZE)

# Constraints
m.addConstr(3*B + 2*G <= 80, name="woodworking")
m.addConstr(2*B + 4*G <= 70, name="nails")
m.addConstr(3*B + 3*G <= 90, name="varnish")

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print(f"Optimal profit: ${m.objVal:.2f}")
    print(f"Bookcases to make: {B.varValue:.2f}")
    print(f"Garden chairs to make: {G.varValue:.2f}")
else:
    print("The problem is infeasible or no solution exists.")
```