Let's break down the problem step by step, using the information and equations provided. --- ## 1. **Variables and Equations** ### **Variables** - **t** = age of the car (1 ≤ t ≤ 5 years) - **V** = volume of engine (100 ≤ V ≤ 250 cc) - **D** = number of doors (2, 3, 4, 5) - **A** = accessories and style (A = 1, 2, 3, 4, 5) - **x** = number of miles driven annually ### **Cost Equations** - **Purchase cost:** \[ \text{Purchase cost} = \left(1 + \frac{t}{5}\right) \times V \times \left(\frac{D}{7} + A\right) \] - **Annual maintenance cost:** \[ \text{Annual maintenance cost} = \frac{t}{2} \times V \times \frac{x}{100} \] - **Annual operating cost:** \[ \text{Annual operating cost} = (D \times t + \frac{V}{100}) \times \frac{x}{20} \] --- ## 2. **Questions and Solutions** --- ### **a. Plot the purchase cost as a function of the four variables** The purchase cost (PC) is: \[ PC = \left(1 + \frac{t}{5}\right) \times V \times \left(\frac{D}{7} + A\right) \] - **t** increases → PC increases - **V** increases → PC increases - **D** increases → PC increases (since \( \frac{D}{7} \) increases) - **A** increases → PC increases **Graph:** You would plot multiple 3D or 2D graphs by fixing two variables and varying the others. For example, plot PC vs V for different t, D, and A. --- ### **b. Which variable has the greatest effect on cost?** **Effect on purchase cost:** - **V (engine volume)** is multiplicative with the other terms, so increases in V will have a large effect. - **A (accessories/style)** is additive to \( \frac{D}{7} \), but as A can go up to 5, it also has a significant impact. - **t (age)** has a smaller effect since it's divided by 5 and added to 1 (i.e., range from 1.2 to 2). - **D (doors)** ranges from 2 to 5, so \( \frac{D}{7} \) ranges from around .29 to .71. **Greatest effect:** **V (engine volume)**, since it directly multiplies the entire expression. --- ### **c. You have a total of $5,000. List the different types of cars (combinations of the parameters) that you can afford.** Set PC ≤ 500: \[ \left(1 + \frac{t}{5}\right) \times V \times \left(\frac{D}{7} + A\right) \leq 500 \] Let's use **minimum values** to find the feasible combinations: - Smallest t = 1 - Smallest V = 100 - Smallest D = 2 - Smallest A = 1 Calculate minimum PC: \[ \left(1 + \frac{1}{5}\right) \times 100 \times \left(\frac{2}{7} + 1\right) = 1.2 \times 100 \times (.2857 + 1) = 1.2 \times 100 \times 1.2857 \approx 1543 \] This is well below $500. Now, **increase variables step by step** to see the limits. Let's try **maximums** (t = 5, V = 250, D = 5, A = 5): \[ \left(1 + \frac{5}{5}\right) \times 250 \times \left(\frac{5}{7} + 5\right) = 2 \times 250 \times (.7143 + 5) = 2 \times 250 \times 5.7143 = 2 \times 14285.75 = 28571.5 \] Way above $500. **To find the combinations**, you would need to solve for each parameter while keeping the others at their minimum, then incrementally increase. The most affordable cars will have: - Smallest t (newest, but still used: 1 year) - Smallest V (100 cc) - Smallest D (2 doors) - Smallest A (1) You can afford cars with slightly higher values of D and A, but not much higher V or t. --- ### **d. Develop a model by which you select the best car for your needs.** - **Define your needs:** (e.g., more doors for family, smaller V for fuel efficiency, higher A for luxury) - **Set a budget constraint:** (e.g., ≤ $500) - **Assign weights to each parameter based on importance** - **Calculate total cost = purchase cost + annual maintenance + annual operating cost** - **Choose the combination that fits your needs and minimizes total cost** --- ### **e. For a person who drives 12,000 miles/year, what is the best car (combination of parameters)?** **Let's minimize total annual operating cost for \( x = 12,000 \):** #### **Calculate all costs for lowest and highest parameter values:** #### **Case 1: Minimum values** - t = 1, V = 100, D = 2, A = 1 1. **Purchase cost:** \[ PC = 1.2 \times 100 \times 1.2857 = 1542.86 \] 2. **Annual maintenance:** \[ = .5 \times 100 \times \frac{12000}{100} = .5 \times 100 \times 120 = 60000 \] 3. **Annual operating:** \[ = (2 \times 1 + \frac{100}{100}) \times \frac{12000}{20} = (2 + 10) \times 600 = 12 \times 600 = 720 \] **Total for 1st year:** \[ \text{Total} = 1542.86 + 60000 + 720 = 68742.86 \] --- #### **Case 2: Max values** - t = 5, V = 250, D = 5, A = 5 1. **Purchase cost:** \[ PC = 2 \times 250 \times 5.7143 = 28571.5 \] 2. **Annual maintenance:** \[ = 2.5 \times 250 \times 120 = 2.5 \times 250 \times 120 = 750 \times 120 = 900000 \] 3. **Annual operating:** \[ = (5 \times 5 + \frac{250}{100}) \times 600 = (25 + 25) \times 600 = 50 \times 600 = 30000 \] **Total for 1st year:** \[ \text{Total} = 28571.5 + 900000 + 30000 = 958571.5 \] --- #### **Minimizing Total Cost** - **Low t, low V, low D, low A** gives the lowest total cost. - **Best combination:** **t = 1, V = 100, D = 2, A = 1** --- ## **Final Answers** ### **a.** The purchase cost is an increasing function of t, V, D, and A. ### **b.** **V (engine volume)** has the greatest effect on cost. ### **c.** You can afford cars with lowest V, t, D, and A (and some with slightly higher D/A). ### **d.** Select the car that meets your needs and minimizes the sum of purchase, maintenance, and operating cost. ### **e.** **For 12,000 miles/year, the best car is:** - **1-year-old** - **100 cc engine** - **2 doors** - **Accessories/style = 1** *This combination yields the lowest total cost for 12,000 miles annually.* --- **If you need a table or a plot, let me know!**