## Step 1: Define the variables and their attributes
We have four variables: 'hours worked by Dale', 'hours worked by Mary', 'hours worked by Jean', and 'hours worked by Peggy'. Let's denote them as $D$, $M$, $J$, and $P$ respectively.

## 2: Define the objective function
The objective function to maximize is $3.77D + 4.37M + 2.05J + 7.17P$.

## 3: Define the constraints
- $D \geq 0$ (non-negative hours worked by Dale, but we are told it can be non-integer)
- $M \geq 0$ and integer (hours worked by Mary)
- $J \geq 0$ and integer (hours worked by Jean)
- $P \geq 0$ and integer (hours worked by Peggy)

## 4: List all given constraints with their mathematical representations
1. $15D + 23M + 19P \geq 96$
2. $15D + 6J + 19P \geq 96$
3. $23M + 6J + 19P \geq 96$
4. $15D + 23M + 19P \geq 93$
5. $15D + 6J + 19P \geq 93$
6. $23M + 6J + 19P \geq 93$
7. $15D + 23M + 19P \geq 105$
8. $15D + 6J + 19P \geq 105$
9. $23M + 6J + 19P \geq 105$
10. $13D + 2P \geq 92$
11. $13D + 11M \geq 141$
12. $15D + 23M \leq 349$
13. $6J + 19P \leq 464$
14. $23M + 19P \leq 446$
15. $15D + 23M + 19P \leq 393$
16. $15D + 23M + 6J \leq 484$
17. $23M + 6J + 19P \leq 176$
18. $15D + 6J + 19P \leq 215$
19. $15D + 23M + 6J + 19P \leq 215$
20. $13D + 20J \leq 533$
21. $20J + 2P \leq 396$
22. $11M + 2P \leq 320$
23. $13D + 11M + 2P \leq 522$
24. $13D + 11M + 20J + 2P \leq 522$

## 5: Implement the problem in Gurobi
We will use Gurobi's Python API to model and solve this problem.

```python
import gurobi as gp

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

# Define variables
D = m.addVar(lb=0, name="D")  # Continuous
M = m.addVar(lb=0, vtype=gp.GRB.INTEGER, name="M")  # Integer
J = m.addVar(lb=0, vtype=gp.GRB.INTEGER, name="J")  # Integer
P = m.addVar(lb=0, vtype=gp.GRB.INTEGER, name="P")  # Integer

# Objective function
m.setObjective(3.77*D + 4.37*M + 2.05*J + 7.17*P, gp.GRB.MAXIMIZE)

# Constraints
m.addConstr(15*D + 23*M + 19*P >= 96)
m.addConstr(15*D + 6*J + 19*P >= 96)
m.addConstr(23*M + 6*J + 19*P >= 96)
m.addConstr(15*D + 23*M + 19*P >= 93)
m.addConstr(15*D + 6*J + 19*P >= 93)
m.addConstr(23*M + 6*J + 19*P >= 93)
m.addConstr(15*D + 23*M + 19*P >= 105)
m.addConstr(15*D + 6*J + 19*P >= 105)
m.addConstr(23*M + 6*J + 19*P >= 105)
m.addConstr(13*D + 2*P >= 92)
m.addConstr(13*D + 11*M >= 141)
m.addConstr(15*D + 23*M <= 349)
m.addConstr(6*J + 19*P <= 464)
m.addConstr(23*M + 19*P <= 446)
m.addConstr(15*D + 23*M + 19*P <= 393)
m.addConstr(15*D + 23*M + 6*J <= 484)
m.addConstr(23*M + 6*J + 19*P <= 176)
m.addConstr(15*D + 6*J + 19*P <= 215)
m.addConstr(15*D + 23*M + 6*J + 19*P <= 215)
m.addConstr(13*D + 20*J <= 533)
m.addConstr(20*J + 2*P <= 396)
m.addConstr(11*M + 2*P <= 320)
m.addConstr(13*D + 11*M + 2*P <= 522)
m.addConstr(13*D + 11*M + 20*J + 2*P <= 522)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("D: ", D.varValue)
    print("M: ", M.varValue)
    print("J: ", J.varValue)
    print("P: ", P.varValue)
else:
    print("The problem is infeasible")
```