Let's solve the problem step by step, with explanations for each part: --- # **Step 1: Understanding the Variables and Equations** ### **Variables:** - \( t \): Age of the car (1 ≤ t ≤ 5 years) - \( V \): Engine volume (100 ≤ V ≤ 250 cc) - \( D \): Number of doors (D = 2, 3, 4, 5) - \( A \): Accessories and style (A = 1, 2, 3, 4, 5) - \( n \): Number of miles driven annually ### **Purchase Cost Equation:** \[ \text{Purchase cost} = \left(1 + \frac{1}{t}\right) \times V \times \left(\frac{D}{7} + A\right) \] ### **Annual Costs:** - **Annual maintenance cost:** \[ = \frac{t}{2} \times V \times \frac{n}{10,000} \] - **Annual operating cost:** \[ = (D + V + \frac{t}{100}) \times \frac{n}{5,000} \] --- # **Step 2: Plot the Purchase Cost as a Function of the Four Variables** Let's break down the purchase cost equation: \[ \text{Purchase cost} = \left(1 + \frac{1}{t}\right) \times V \times \left(\frac{D}{7} + A\right) \] - **As \( t \) increases** (older car), \( \frac{1}{t} \) decreases, so the multiplier decreases. - **As \( V \) increases** (bigger engine), cost increases proportionally. - **As \( D \) increases**, \( \frac{D}{7} \) increases, so cost increases. - **As \( A \) increases**, cost increases. **Graphically:** Cost increases with higher \( V \), \( D \), or \( A \), but decreases as \( t \) increases. --- # **Step 3: Which Variable Has the Greatest Effect on Cost?** Let’s examine the purchase cost equation: - **\( V \)** is a direct multiplier; increasing \( V \) by 2x increases cost by 2x. - **\( t \)** is in the denominator inside \( (1 + \frac{1}{t}) \). Lower \( t \) (newer car) increases cost sharply. - **\( D \)** and **\( A \)** are inside the additive group, but \( A \) can be up to 5, while \( D/7 \) ranges from ~.29 to ~.71. **Conclusion:** - **Engine volume \( V \)** has the strongest direct impact on purchase cost. - **Age \( t \)** also has a significant effect, especially for newer cars. - **Accessories \( A \)** has a moderate effect. - **Doors \( D \)** has the smallest effect. --- # **Step 4: Efficient Choices for a $5,000 Budget** You want to maximize the value for $5,000 or less. \[ \left(1 + \frac{1}{t}\right) \times V \times \left(\frac{D}{7} + A\right) \leq 5,000 \] - **To minimize cost:** Choose higher \( t \), lower \( V \), lower \( D \), and lower \( A \). - **Specifics:** - **Oldest car (t = 5)** - **Smallest engine (V = 100 cc)** - **Fewest doors (D = 2)** - **Least accessories (A = 1)** \[ \left(1 + \frac{1}{5}\right) \times 100 \times \left(\frac{2}{7} + 1\right) = 1.2 \times 100 \times (.286 + 1) = 1.2 \times 100 \times 1.286 = 1.2 \times 1286 = \approx \$1543 \] So plenty of room under $5,000 for these specs. You can adjust \( D \) or \( A \) upward for more features if desired. --- # **Step 5: Model for Selecting the Best Car** **Criteria:** - **Maximize utility** (e.g., more doors for family, more accessories for comfort) - **Minimize cost** (purchase + annual costs) **Model:** 1. List all possible combinations meeting your needs. 2. Calculate total cost (purchase + annual maintenance + annual operating): \[ \text{Total cost in a year} = \text{Purchase cost} + \text{Annual maintenance} + \text{Annual operating cost} \] 3. Select the combination with the best balance of features and lowest total cost. --- # **Step 6: Best Car for Driving 12,000 Miles Annually** - \( n = 12,000 \) - Use the cheapest possible car (from above), or adjust for your needs. **Annual maintenance:** \[ = \frac{t}{2} \times V \times \frac{n}{10,000} \] \[ = \frac{5}{2} \times 100 \times \frac{12,000}{10,000} = 2.5 \times 100 \times 1.2 = 250 \times 1.2 = \$300 \] **Annual operating:** \[ = (D + V + \frac{t}{100}) \times \frac{n}{5,000} \] \[ = (2 + 100 + .05) \times \frac{12,000}{5,000} \] \[ = (1002.05) \times 2.4 = \$2404.92 \] **Total annual cost:** \[ = \$300 + \$2404.92 = \$5404.92 \] **+ initial purchase cost (\$1543)** --- # **Final Conclusion** - **Best car for a low budget:** Oldest, smallest engine, fewest doors, least accessories. - **Total cost (first year):** \$1543 (purchase) + \$5404.92 (annual costs) ≈ **\$6947.92** - **Variables with the most impact:** Engine volume \( V \) and car age \( t \). - **For different needs:** Increase \( D \) or \( A \) as required, but this increases cost. **Recommendation:** For someone driving 12,000 miles/year on a $5,000 budget, the best choice is the oldest model (5 years), smallest engine (100cc), 2 doors, and minimal accessories (A=1). Adjust upward for more features as budget allows. --- **Let me know if you want to see a graph or table for these variables!**

# Step-by-Step Solution to the Car Cost Problem ## Step 1: Understanding the Variables and Equations ### Variables - **\( t \)**: Age of the car (1 to 5 years) - **\( V \)**: Engine volume (100 to 250 cc) - **\( D \)**: Number of doors (2, 3, 4, 5) - **\( A \)**: Accessories and style (1 to 5) - **\( n \)**: Number of miles driven annually ### Equations - **Purchase Cost** \[ \text{Purchase cost} = \left(1 + \frac{1}{t}\right) \times V \times \left(\frac{D}{7} + A\right) \] - **Annual Maintenance Cost** \[ = \frac{t}{2} \times V \times \frac{n}{10,000} \] - **Annual Operating Cost** \[ = (D + V + \frac{t}{100}) \times \frac{n}{5,000} \] **Explanation:** Understanding the variables and equations is crucial as they define how the purchase and maintenance costs of a car are calculated. Each variable influences the cost in different ways, which will help in making informed decisions. --- ## Step 2: Plot the Purchase Cost as a Function of the Four Variables ### Analysis of Purchase Cost Equation \[ \text{Purchase cost} = \left(1 + \frac{1}{t}\right) \times V \times \left(\frac{D}{7} + A\right) \] - **Effect of \( t \)**: As \( t \) increases (the car gets older), the multiplier \( \frac{1}{t} \) decreases, lowering the cost. - **Effect of \( V \)**: Increasing \( V \) increases the cost proportionally. - **Effect of \( D \)**: Increasing \( D \) increases cost via \( \frac{D}{7} \). - **Effect of \( A \)**: Increasing \( A \) directly increases the cost. **Explanation:** This step highlights how the purchase cost fluctuates based on each variable. Understanding these relationships allows for the optimization of costs based on personal needs. --- ## Step 3: Which Variable Has the Greatest Effect on Cost? ### Comparison of Variables - **Engine Volume \( V \)**: Directly affects the cost; doubling \( V \) doubles the cost. - **Age \( t \)**: Newer cars (lower \( t \)) significantly increase costs due to \( \frac{1}{t} \). - **Doors \( D \)** and **Accessories \( A \)**: Have a smaller impact compared to \( V \) and \( t \). **Conclusion:** The most significant factors affecting purchase cost are engine volume \( V \) and car age \( t \). Accessories \( A \) and doors \( D \) have a lesser impact. **Explanation:** Identifying which variables have the greatest effect on cost is essential for making budget-conscious choices. This analysis allows prioritization of features based on their financial implications. --- ## Step 4: Efficient Choices for a $5,000 Budget ### Budget Constraint \[ \left(1 + \frac{1}{t}\right) \times V \times \left(\frac{D}{7} + A\right) \leq 5,000 \] ### Optimal Choices - **Maximize \( t \)**: Choose \( t = 5 \) (oldest car) - **Minimize \( V \)**: Choose \( V = 100 \) cc - **Minimize \( D \)**: Choose \( D = 2 \) - **Minimize \( A \)**: Choose \( A = 1 \) **Calculation of Purchase Cost:** \[ \left(1 + \frac{1}{5}\right) \times 100 \times \left(\frac{2}{7} + 1\right) \approx 1543 \] **Explanation:** This step outlines how to maximize value while staying within budget. By selecting the lowest values for \( V \), \( D \), and \( A \), you ensure the purchase cost remains manageable. --- ## Step 5: Model for Selecting the Best Car ### Criteria for Selection 1. **Maximize Utility**: Consider family needs, comfort. 2. **Minimize Costs**: Include purchase, maintenance, and operational costs. ### Calculation of Total Cost \[ \text{Total cost in a year} = \text{Purchase cost} + \text{Annual maintenance} + \text{Annual operating cost} \] **Explanation:** This step emphasizes the importance of considering both initial and ongoing costs. A comprehensive evaluation allows for more informed decisions regarding car purchases. --- ## Step 6: Best Car for Driving 12,000 Miles Annually ### Annual Cost Calculations - **Annual Maintenance Cost** \[ = \frac{t}{2} \times V \times \frac{12,000}{10,000} \approx 300 \] - **Annual Operating Cost** \[ = (D + V + \frac{t}{100}) \times \frac{12,000}{5,000} \approx 2404.92 \] ### Total Annual Cost \[ = 1543 + 5404.92 \approx 6947.92 \] **Explanation:** This step calculates the total cost of ownership over a year, giving a clear picture of financial obligations. Understanding these costs aids in making the best car choice for annual driving needs. --- # Final Conclusion - **Best option for low budget**: The oldest car, smallest engine, fewest doors, and minimal accessories. - **Total first-year cost**: Approximately \$6947.92. - **Significant variables**: Engine volume \( V \) and car age \( t \) are crucial for cost management. **Recommendation:** Choose a car tailored to the driving needs while considering how features affect overall costs. Analyzing these factors facilitates a balance between budget and utility.