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) \(22a + 35b \leq 4500\) \(a + b \geq 100\) \(a \geq 20\) \(b \geq 15\)