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 home vacuums repaired,
- $x_2$ as the number of shop vacuums repaired.

The profit per home vacuum repaired is $20, and the profit per shop vacuum repaired is $35. Therefore, the objective function to maximize profit can be represented as:
\[ \text{Maximize:} \quad 20x_1 + 35x_2 \]

Given the constraints:
- Each shop vacuum requires 1 hour of disassembly and each home vacuum requires 0.5 hours of disassembly, with a total of 300 hours available for disassembly.
- Each shop vacuum requires 2 hours of repair and each home vacuum requires 1 hour of repair, with a total of 400 hours available for repair.

The constraints can be represented as:
\[ 0.5x_1 + x_2 \leq 300 \] (disassembly time constraint)
\[ x_1 + 2x_2 \leq 400 \] (repair time constraint)

Also, the number of vacuums cannot be negative, so we have:
\[ x_1 \geq 0 \]
\[ x_2 \geq 0 \]

In symbolic representation, this problem can be summarized as:

```json
{
    'sym_variables': [('x1', 'number of home vacuums repaired'), ('x2', 'number of shop vacuums repaired')],
    'objective_function': '20*x1 + 35*x2',
    'constraints': ['0.5*x1 + x2 <= 300', 'x1 + 2*x2 <= 400', 'x1 >= 0', 'x2 >= 0']
}
```

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

```python
from gurobipy import *

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

# Define variables
x1 = m.addVar(name="home_vacuums", vtype=GRB.CONTINUOUS, lb=0)
x2 = m.addVar(name="shop_vacuums", vtype=GRB.CONTINUOUS, lb=0)

# Set the objective function
m.setObjective(20*x1 + 35*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(0.5*x1 + x2 <= 300, name="disassembly_time")
m.addConstr(x1 + 2*x2 <= 400, name="repair_time")

# Optimize the model
m.optimize()

# Print the results
for v in m.getVars():
    print(f"{v.varName}: {v.x}")

print(f"Objective: {m.objVal}")
```