## Symbolic Representation

To solve the given problem, we first need to convert the natural language description into a symbolic representation. Let's denote:

- \(x_1\) as the number of standard definition web-cams
- \(x_2\) as the number of high definition web-cams

The objective is to maximize profit. The profit from selling \(x_1\) standard definition web-cams is $100\(x_1\) and from selling \(x_2\) high definition web-cams is $125\(x_2\). Therefore, the objective function to maximize is:

\[ \text{Maximize:} \quad 100x_1 + 125x_2 \]

The constraints given are:

1. The total cost of the web-cams should not exceed $40,000. The cost for \(x_1\) standard definition web-cams is $150\(x_1\) and for \(x_2\) high definition web-cams is $250\(x_2\). So, the first constraint is:

\[ 150x_1 + 250x_2 \leq 40000 \]

2. The total number of web-cams should not exceed 275:

\[ x_1 + x_2 \leq 275 \]

3. Non-negativity constraints, as the number of web-cams cannot be negative:

\[ x_1 \geq 0, \quad x_2 \geq 0 \]

## Symbolic Representation in JSON Format

```json
{
    'sym_variables': [('x1', 'standard definition web-cams'), ('x2', 'high definition web-cams')],
    'objective_function': '100*x1 + 125*x2',
    'constraints': [
        '150*x1 + 250*x2 <= 40000',
        'x1 + x2 <= 275',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

## Gurobi Code in Python

```python
import gurobipy as gp

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

# Define variables
x1 = model.addVar(name="standard_definition_webcams", lb=0, vtype=gp.GRB.INTEGER)
x2 = model.addVar(name="high_definition_webcams", lb=0, vtype=gp.GRB.INTEGER)

# Objective function: Maximize profit
model.setObjective(100*x1 + 125*x2, gp.GRB.MAXIMIZE)

# Constraints
model.addConstr(150*x1 + 250*x2 <= 40000, name="inventory_cost_constraint")
model.addConstr(x1 + x2 <= 275, name="demand_constraint")

# Solve the model
model.optimize()

# Print the solution
if model.status == gp.GRB.OPTIMAL:
    print("Optimal Solution:")
    print(f"Standard Definition Webcams: {x1.varValue}")
    print(f"High Definition Webcams: {x2.varValue}")
    print(f"Max Profit: {model.objVal}")
else:
    print("The model is infeasible or unbounded.")
```