Let’s solve the problem mathematically.

Step 1: Define Variables

• Let $x$ = amount invested in stocks.
• Let $y$ = amount invested in bonds.
• Let $z$ = amount invested in money market funds.

Step 2: Constraints and Objective

Constraints:

1. Total investment cannot exceed $300,000: $x + y + z \leq 300,000$
2. The amount invested in bonds must be at least equal to the sum of stocks and money market funds: $y \geq x + z$
3. At least $75,000 must be invested in stocks and money market funds: $x + z \geq 75,000$

Objective:

Maximize the annual return: $\text{Return} = 0.15x + 0.10y + 0.08z$

Step 3: Solve the Problem

This is a linear programming problem. I will compute the solution to maximize the return based on the constraints.from scipy.optimize import linprog

Coefficients of the objective function (maximize return, so negative for linprog)

c = [-0.15, -0.10, -0.08]

Coefficients for inequality constraints

A = [ [1, 1, 1], # Total investment ≤ 300,000 [-1, 1, -1], # y ≥ x + z -> -x + y - z ≥ 0 [-1, 0, -1] # x + z ≥ 75,000 -> -x - z ≤ -75,000 ]

Right-hand side of inequality constraints

b = [300000, 0, -75000]

Bounds for each variable (all investments must be non-negative)

x_bounds = (0, None) # x ≥ 0 y_bounds = (0, None) # y ≥ 0 z_bounds = (0, None) # z ≥ 0

Solve the linear programming problem

result = linprog(c, A_ub=A, b_ub=b, bounds=[x_bounds, y_bounds, z_bounds], method='highs')

Extract results

result_values = result.x # Investments in stocks, bonds, money market max_return = -result.fun # Maximized return

result_values, max_return(array([300000., 0., 0.]), 45000.0)(array([300000., 0., 0.]), 45000.0)successSandra should invest $300,000 in stocks, $0 in bonds, and $0 in money market funds to maximize her return. This allocation provides an annual return of $45,000.

Let me know if you need further clarification or a detailed breakdown!

Here are some related questions to consider:

1. How would the solution change if Sandra wanted at least $100,000 invested in bonds?
2. What if the annual yields for these investments changed?
3. Could Sandra diversify her investments further and still maximize her return?
4. How would taxes or fees on returns affect the investment strategy?
5. What is the impact of reducing the total available capital to $200,000?

Tip: Always check if constraints in real-world problems are flexible; adjusting them slightly could often lead to better results.