Question

Heinrich must buy at least 100 shares of stock for his portfolio. The shares he buys will be from Stock \(X\), which costs \(\$ 22\) per share and Stock \(Y\), which costs \(\$ 35\) per share. His budget for buying stock is no more than \(\$ 4,500\). He must buy at least 20 shares of Stock \(X\) and 15 shares of Stock Y. Which of the following represents the situation described if \(a\) is the number of shares of Stock X purchased and \(b\) is the number of shares of Stock \(Y\) purchased? A) \(\begin{aligned} & 22 a+35 b \leq \\ & a+b \geq 100 \\ & a \leq 20 \\ & b \leq 15\end{aligned}\) B) \(22 a+35 b \leq 4,500\) \(a+b \leq 100\) \(a \leq 20\) \(b \leq 15\) C) \(22 a+35 b \leq 4,500\) \(a+b \leq 100\) \(a \geq 20\) \(b \geq 15\) D) \(22 a+35 b \leq 4,500\) \(a+b \geq 100\) \(a \geq 20\) \(b \geq 15\) $$ x^2-8 x+5 $$

Short answer

The short answer is: D) \(22a + 35b \leq 4500\) \(a + b \geq 100\) \(a \geq 20\) \(b \geq 15\)

Step-by-step solution

Option A Analysis:

Option A: 1) \( 22a + 35b \leq \) 2) \( a + b \geq 100\) 3) \( a \leq 20\) 4) \( b \leq 15\) Notice in Option A, the budget constraint is not given. Since the constraint is crucial, we can rule out Option A.

Option B Analysis:

Option B: 1) \(22a + 35b \leq 4500\) 2) \(a + b \leq 100\) 3) \(a \leq 20\) 4) \(b \leq 15\) Notice in Option B, the minimums for Stock X and Y are represented as maximums (i.e., \(a \leq 20\) and \(b \leq 15\)). Since this contradicts the information given, we can rule out Option B.

Option C Analysis:

Option C: 1) \(22a + 35b \leq 4500\) 2) \(a + b \leq 100\) 3) \(a \geq 20\) 4) \(b \geq 15\) Notice in Option C, the constraint requiring at least 100 total shares of stock is given as a maximum constraint (\(a+b \leq 100\)). This contradicts the information given, so we can rule out Option C.

Option D Analysis:

Option D: 1) \(22a + 35b \leq 4500\) 2) \(a + b \geq 100\) 3) \(a \geq 20\) 4) \(b \geq 15\) Option D accurately represents the budget constraint, the minimum shares of each stock, and the overall minimum shares of stock for the portfolio. Therefore, the correct answer is: D) \(22a + 35b \leq 4500\) \(a + b \geq 100\) \(a \geq 20\) \(b \geq 15\)

Key concepts

Mathematical Modeling

Mathematical modeling involves using mathematical expressions to represent real-world situations. In this exercise, we represent Heinrich's investment constraints using linear inequalities. Each inequality corresponds to a different aspect of the problem: budget limitation, total stock requirement, and minimum purchase quantities. Linear inequalities are essential in forming systems that help us make informed decisions based on given limits. For Heinrich, two variables, \(a\) and \(b\), represent the number of shares for Stock X and Stock Y, respectively. The inequalities are constructed to ensure all conditions are accurately obeyed in mathematical form. This way, the model provides a clear, logical framework to approach similar issues in portfolio management.

Budget Constraints

Budget constraints limit the amount of money available for spending and are a crucial aspect of financial planning. Here, Heinrich has a budget of no more than \( \$4500 \) to purchase stocks. The equation \( 22a + 35b \leq 4500 \) reflects his financial limitation where \(22\) and \(35\) are the costs per share of Stocks X and Y.

By setting up this constraint, Heinrich ensures that his spending does not exceed his designated financial limits. Budget constraints are vital in ensuring that financial resources are wisely allocated and that any investment plan remains financially feasible. They guide diligent financial discipline, preventing overspend that can otherwise lead to financial distress.

Portfolio Management

Portfolio management is the art of selecting the right combination of investments to achieve specific objectives. Heinrich aims to construct a portfolio with at least 100 shares while making informed decisions about his stock choices, adhering to certain constraints.

The exercise illustrates important aspects of portfolio management: diversification, maximum allocation for each asset, and adhering to a total share target. In real scenarios, other factors like risk tolerance and expected returns are also considered. The goal is to balance these factors to achieve Heinrich's investment strategies within acceptable risk parameters, showcasing the strategic nature of effective portfolio management.

Inequality Constraints

Inequality constraints are conditions that specify the range within which a solution should lie. In mathematical programming, they form the boundaries that limit possible solutions in optimization problems.

For Heinrich, inequality constraints ensure he meets his minimum required purchases for each stock: at least 20 shares of Stock X and 15 shares of Stock Y, symbolized by \( a \geq 20 \) and \( b \geq 15 \).

• These inequalities guide decision-making by clearly indicating constraints on share purchases.
• They ensure operational feasibility by maintaining conditions of the financial model.

By systematically analyzing these constraints, Heinrich strategically navigates his investment decisions, highlighting the importance of methodically considering all limitations in financial modeling.