## Step 1: Define the symbolic variables
Let's define the symbolic variables for the problem:
- $x_1$ represents the acres of carrots
- $x_2$ represents the acres of onions

## Step 2: Formulate the objective function
The profit per acre of carrots is $75, and the profit per acre of onions is $90. The objective function to maximize profit is:
\[ \text{Maximize:} \quad 75x_1 + 90x_2 \]

## 3: Define the constraints
1. Total land constraint: Lucas has 120 acres of land.
\[ x_1 + x_2 \leq 120 \]
2. Tractor time constraint: There are 120 days available for tractor use. It takes 1.5 days of tractor time per acre of carrots and 2 days per acre of onions.
\[ 1.5x_1 + 2x_2 \leq 120 \]
3. Hand-picking time constraint: There are 200 days available for hand-picking. It takes 2.5 days per acre of carrots and 2 days per acre of onions.
\[ 2.5x_1 + 2x_2 \leq 200 \]
4. Non-negativity constraints: Acres of land cannot be negative.
\[ x_1 \geq 0, x_2 \geq 0 \]

## 4: Symbolic representation of the problem
```json
{
'sym_variables': [('x1', 'acres of carrots'), ('x2', 'acres of onions')],
'objective_function': '75*x1 + 90*x2',
'constraints': [
'x1 + x2 <= 120',
'1.5*x1 + 2*x2 <= 120',
'2.5*x1 + 2*x2 <= 200',
'x1 >= 0',
'x2 >= 0'
]
}
```

## 5: Gurobi code to solve the problem
```python
import gurobi

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

    # Define the variables
    x1 = model.addVar(lb=0, name="carrots")
    x2 = model.addVar(lb=0, name="onions")

    # Define the objective function
    model.setObjective(75 * x1 + 90 * x2, gurobi.GRB.MAXIMIZE)

    # Add constraints
    model.addConstr(x1 + x2 <= 120, name="land_constraint")
    model.addConstr(1.5 * x1 + 2 * x2 <= 120, name="tractor_constraint")
    model.addConstr(2.5 * x1 + 2 * x2 <= 200, name="hand_picking_constraint")

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Acres of carrots: {x1.varValue}")
        print(f"Acres of onions: {x2.varValue}")
        print(f"Maximum profit: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```