## Step 1: Define the variables and the objective function
The problem involves maximizing an objective function with variables representing the hours worked by Laura, Ringo, John, Paul, Hank, and George. Let's denote these variables as $L$, $R$, $J$, $P$, $H$, and $G$ respectively.

## 2: Formulate the objective function
The objective function to maximize is:
\[ 9.65LR + 7.47LJ + 6.82LH + 6.51LG + 7.67R^2 + 3.64RH + 1.08RG + 9.24J^2 + 9.39JH + 1.46JG + 8.24P^2 + 7.27PG + 9.2H^2 + 6.82HG + 1.5G^2 + 9.53R + 1.88J + 8.03H + 6.24G \]

## 3: Define the constraints
1. $L = 6$
2. $R = 9$
3. $J = 11$
4. $P = 1$
5. $H = 7$
6. $G = 10$
7. $R^2 + P^2 + G^2 \geq 15$
8. $L + J + H \geq 15$
9. $L^2 + R^2 + P^2 \geq 15$
10. $R^2 + P^2 + G^2 \geq 18$
11. $L + J + H \geq 18$
12. $L^2 + R^2 + P^2 \geq 18$
13. $R^2 + P^2 + G^2 \geq 10$
14. $L + J + H \geq 10$
15. $L^2 + R^2 + P^2 \geq 10$
16. $6J - 3G \geq 0$
17. $R^2 + P^2 \leq 47$
18. $R^2 + G^2 \leq 63$
19. $L + G \leq 56$
20. $H + G \leq 111$
21. $L^2 + R^2 \leq 62$
22. $L + J \leq 110$
23. $J + P \leq 28$
24. $R + J + P \leq 97$
25. $L + P + H \leq 74$
26. $L + R + J + P + H + G \leq 74$
And the variable constraints:
- $L$ is an integer
- $R$ is an integer
- $J$ is continuous
- $P$ is an integer
- $H$ is an integer
- $G$ is an integer

## 4: Implement the problem in Gurobi
Given the complexity and the specific requirements of the problem, we will use Gurobi to solve it. However, constraints 1 through 6 fix the values of $L, R, J, P, H,$ and $G$, which simplifies the problem significantly.

```python
import gurobi as gp

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

# Define the variables
L = m.addVar(lb=6, ub=6, vtype=gp.GRB.INTEGER, name="L")
R = m.addVar(lb=9, ub=9, vtype=gp.GRB.INTEGER, name="R")
J = m.addVar(lb=-gp.GRB.INFINITY, ub=gp.GRB.INFINITY, vtype=gp.GRB.CONTINUOUS, name="J")
P = m.addVar(lb=1, ub=1, vtype=gp.GRB.INTEGER, name="P")
H = m.addVar(lb=7, ub=7, vtype=gp.GRB.INTEGER, name="H")
G = m.addVar(lb=10, ub=10, vtype=gp.GRB.INTEGER, name="G")

# Objective function
m.setObjective(9.65*L*R + 7.47*L*J + 6.82*L*H + 6.51*L*G + 
               7.67*R**2 + 3.64*R*H + 1.08*R*G + 9.24*J**2 + 
               9.39*J*H + 1.46*J*G + 8.24*P**2 + 7.27*P*G + 
               9.2*H**2 + 6.82*H*G + 1.5*G**2 + 9.53*R + 
               1.88*J + 8.03*H + 6.24*G, gp.GRB.MAXIMIZE)

# Constraints
m.addConstr(R**2 + P**2 + G**2 >= 15, name="c7")
m.addConstr(L + J + H >= 15, name="c8")
m.addConstr(L**2 + R**2 + P**2 >= 15, name="c9")
m.addConstr(R**2 + P**2 + G**2 >= 18, name="c10")
m.addConstr(L + J + H >= 18, name="c11")
m.addConstr(L**2 + R**2 + P**2 >= 18, name="c12")
m.addConstr(R**2 + P**2 + G**2 >= 10, name="c13")
m.addConstr(L + J + H >= 10, name="c14")
m.addConstr(L**2 + R**2 + P**2 >= 10, name="c15")
m.addConstr(6*J - 3*G >= 0, name="c16")
m.addConstr(R**2 + P**2 <= 47, name="c17")
m.addConstr(R**2 + G**2 <= 63, name="c18")
m.addConstr(L + G <= 56, name="c19")
m.addConstr(H + G <= 111, name="c20")
m.addConstr(L**2 + R**2 <= 62, name="c21")
m.addConstr(L + J <= 110, name="c22")
m.addConstr(J + P <= 28, name="c23")
m.addConstr(R + J + P <= 97, name="c24")
m.addConstr(L + P + H <= 74, name="c25")
m.addConstr(L + R + J + P + H + G <= 74, name="c26")

# Optimize the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("L: ", L.varValue)
    print("R: ", R.varValue)
    print("J: ", J.varValue)
    print("P: ", P.varValue)
    print("H: ", H.varValue)
    print("G: ", G.varValue)
else:
    print("The model is infeasible")
```