Let's break down the **questions** and the **context**: --- ### **Given:** - **Decision variables:** \(X_1\) is the number of City Democrats interviewed, etc. - **Costs:** - **City:** Democrat $18, Republican $16, Independent $22 - **Suburbs:** Democrat $21, Republican $20, Independent $25 --- #### **Question 12:** > At the optimal solution, which statement is CORRECT about \(X_1\) (number of City Democrats interviewed)? Options: - a. \(100 < X_1 \leq 200\) - b. \(200 < X_1 \leq 300\) - c. \(500 < X_1 \leq 600\) - d. \(400 < X_1 \leq 500\) - e. \(X_1 > 600\) - f. \(X_1 \leq 100\) - g. \(300 < X_1 \leq 400\) #### **Question 13:** > Which statement is CORRECT about the optimal objective function value \(Z\)? Options: - a. $25,000 < Z \leq $30,000 - b. \(Z > $30,000\) - c. $20,000 < Z \leq $25,000 - d. $10,000 < Z \leq $15,000 - e. $15,000 < Z \leq $20,000 - f. \(Z \leq $10,000\) --- ### **How to approach (without Excel):** - The actual numbers for \(X_1\) and \(Z\) depend on Excel Solver's results based on constraints (not shown in the image). - **If you already ran Solver and saw the answers, choose the correct range.** --- ### **If you need to make an educated guess:** - **City Democrat** interviews cost $18 (lower than Suburbs Democrat at $21), so likely more interviews will be allocated here for cost efficiency. - **Typical survey sizes** for marketing research tend to be in the hundreds per group. #### **Question 12 - Likely answer:** - The answer should be in a mid-to-high range (not too low, not too high). - Most likely: **d. 400 < X₁ ≤ 500** or **g. 300 < X₁ ≤ 400** #### **Question 13 - Likely answer:** - Total cost will depend on the number of interviews (likely several hundred to a couple thousand) times $15-$25 each. - If you have, for example, 1000 interviews at $20 each = $20,000. - Most likely: **e. $15,000 < Z ≤ $20,000** or **c. $20,000 < Z ≤ $25,000** --- ### **Final Answers (GUESS based on typical values):** - **Question 12:** **d. 400 < X₁ ≤ 500** - **Question 13:** **e. $15,000 < Z ≤ $20,000** --- **If you have the actual Solver output, use those numbers to select the correct option!** If not, these choices are the most reasonable based on standard marketing survey sizes and costs.

Certainly! Here's a step-by-step approach to analyze and answer the questions based on the provided scenario, constraints, and typical linear programming logic. --- ### **Step 1: Understand the context and decision variables** - The company is interviewing different voter groups: - City Democrats (\(X_1\)) - City Republicans (\(X_2\)) - City Independents (\(X_3\)) - Suburbs Democrats (\(X_4\)) - Suburbs Republicans (\(X_5\)) - Suburbs Independents (\(X_6\)) - Costs are associated with each group, and the goal is to minimize total interview costs while satisfying certain constraints (not fully provided here). --- ### **Step 2: Identify the nature of the problem** - The problem is a linear programming (LP) model aimed at cost minimization. - The optimal solution balances the number of interviews across groups to minimize total cost, given constraints on total voters, demographic quotas, or other constraints. --- ### **Step 3: Use intuition and typical LP solutions** - **Cost efficiency:** City Democrat interviews cost $18, which is less than Suburbs Democrat at $21, and much less than Independent interviews in either location. - **Likely allocation:** The LP will favor allocating as many interviews as possible to the lowest-cost groups (City Democrats, then City Republicans, etc.) within the constraints. --- ### **Step 4: Address Question 12** **Question 12:** *"At the optimal solution, which statement is correct about \(X_1\) (City Democrats interviewed)?"* - Since City Democrats are the least costly among Democrat groups and assuming constraints are not extremely restrictive, the LP would allocate a significant number of City Democrat interviews. - Typical sizes for survey samples in marketing research are often in the hundreds, sometimes approaching a few thousand depending on the scope. **Options:** - a. \(100 < X_1 \leq 200\) - b. \(200 < X_1 \leq 300\) - c. \(500 < X_1 \leq 600\) - d. \(400 < X_1 \leq 500\) - e. \(X_1 > 600\) - f. \(X_1 \leq 100\) - g. \(300 < X_1 \leq 400\) **Reasoning:** - Given typical survey sizes and cost considerations, the number of City Democrat interviews is likely in the 400-500 range to optimize costs while meeting demographic needs. **Most reasonable choice:** **d. 400 < X₁ ≤ 500** --- ### **Step 5: Address Question 13** **Question 13:** *"Which statement is correct about the optimal objective function value \(Z\)?"* - The total cost \(Z\) depends on the number of interviews and their respective costs. - Assume, for example, around 400-500 interviews at an average cost of about $20 per interview: \[ Z \approx 400 \times 20 = 8,000 \quad \text{to} \quad 500 \times 20 = 10,000 \] - But the problem likely involves interviewing multiple groups, possibly leading to total costs in the range of $15,000 to $25,000. **Options:** - a. $25,000 < Z ≤ $30,000 - b. \(Z > $30,000\) - c. $20,000 < Z ≤ $25,000 - d. $10,000 < Z ≤ $15,000 - e. $15,000 < Z ≤ $20,000 - f. \(Z \leq $10,000\) **Reasoning:** - Given typical survey sizes and costs, the total cost is likely between $15,000 and $20,000, especially if multiple groups are sampled. **Most reasonable choice:** **e. $15,000 < Z ≤ $20,000** --- ### **Step 6: Final conclusion** - **Question 12:** The number of City Democrat interviews (\(X_1\)) is likely **just under or around 500**. - **Question 13:** The total cost (\(Z\)) is likely **between $15,000 and $20,000**. --- ### **Summary:** | Question | Likely Answer | Explanation | |------------|----------------|--------------| | 12 | **d. 400 < X₁ ≤ 500** | Cost-efficient allocation suggests a large sample in this range. | | 13 | **e. $15,000 < Z ≤ $20,000** | Based on typical interview costs and sample sizes. | --- ### **Note:** - Precise answers depend on the actual solver output and constraints, which are not provided here. - If you have the solver results, verify the exact value of \(X_1\) and total \(Z\) to select the most accurate options. --- **Hope this detailed reasoning helps you understand how to approach such LP problems!**