To solve the optimization problem described, we first need to convert the natural language description into a symbolic representation. This involves defining variables, an objective function, and constraints based on the given information.

Let's define:
- $x_1$ as the number of standard definition web-cams to be stocked.
- $x_2$ as the number of high definition web-cams to be stocked.

The profit from each standard definition web-cam is $100, and from each high definition web-cam is $125. Therefore, the objective function to maximize profit can be written as:
\[ \text{Maximize:} \quad 100x_1 + 125x_2 \]

Given constraints are:
1. The total investment should not exceed $40,000. Since a standard definition web-cam costs $150 and a high definition web-cam costs $250, we have:
\[ 150x_1 + 250x_2 \leq 40000 \]
2. The total monthly demand for web-cams is at most 275, so:
\[ x_1 + x_2 \leq 275 \]
3. Non-negativity constraints since the number of web-cams cannot be negative:
\[ x_1 \geq 0 \]
\[ x_2 \geq 0 \]

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

To solve this linear programming problem using Gurobi in Python, we can use the following code:

```python
from gurobipy import *

# Create a new model
m = Model("WebCam_Optimization")

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

# Set the objective function
m.setObjective(100*x1 + 125*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(150*x1 + 250*x2 <= 40000, "investment_constraint")
m.addConstr(x1 + x2 <= 275, "demand_constraint")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Number of standard definition web-cams: {x1.x}")
    print(f"Number of high definition web-cams: {x2.x}")
    print(f"Maximum profit: {m.objVal}")
else:
    print("No optimal solution found.")
```