Please check my answer and the table and let me know if you think that it is correct.
Maintenance costs include some things that remain constant every year, like oil changes, and others that have a tendency to rise as cars get older. In fact, it's this annual rise in maintenance costs that makes them an essential element in our model. Pause to consider what would happen if the cost of maintaining your car didn't change. Without doing any math, just thinking about it.... maintenance How often should your replace your car if its maintenance costs are the same every year as long as you own it? Explain briefly. Indefinitely , b/ C as long as it runs, the same ( constant amount of If is required to maintain it every year Let's assume that annual maintenance costs rise linearly. That is, they go up by the same amount each year. In fact, let's assume that maintenance costs are $200 the first year, then $400 the second year, $600 the third year, and so on. Complete the table below to determine how much you'll spend each yr year over the first five years of ownership, and also how much total you'll have spent total on maintenance throughout the first year, the first two years, the first three years, the first four years, and the first five years of car ownership. As before, a column is provided for you to write words or expressions that may help you generalize later. Year of Maintenance cost this year Total maintenance costs up How determined? ownership to and including this year. (cumulative costs) $ 200 $ 200 Initial cost / in given 2 400 60 0 200 + 400 = 600 3 600 1200 yr. 2 yr 3 400 + 600 = 1200 4 800 2000 200 + 800 5 1000 3,000 2000 + 1000 = 300 Total Cost of One Car We're ready to put together the essential elements of our model. Let's assume that you own your car for five years. Fill in the blanks below with values you were given or determined earlier. Add them up to determine how much you'll have spent on your car over five years if you replace it after five years. Total Cost = 25,000 8 904, 81 +23,000. cc purchase price resale value cumulative maintenance Total Lifetime Cost of Car Ownership You've now estimated how much it will cost you to own a car for five years. But our goal was to determine the cost of many cars over your lifetime. Let's assume you end up driving a car for 50 years. If you replace your car every five years over 50 years of driving, and if each car costs you the amount that you wrote at the bottom of the previous page, determine how much you'll spend on car ownership over your entire lifetime. Include notes to yourself about how you determined this. yrs / car 50. + 5. = 10 cars 19, 095 , 19 x 10 cars $ 190, 951. 90 lifetime for 10 cars A General Model Up to now we've been assuming that you replaced your car every 5 years. But our actual goal is to determine how often you should replace your car if you're trying to minimize the lifetime cost. We can't tell yet, because we only know the lifetime cost if you replace your car every 5 years. What if you replace it every 8 years? Every 2 years? Should we go back and redo all the previous work using these different numbers? We could, but that's not the way mathematical modeling works. Instead, we want to generalize, coming up with an equation (that's the model) for the lifetime cost of car ownership if you replace your car every x years. We'll keep the assumption that the purchase price is $25,000. But we need to generalize the resale value. Look back at your table on page 2 of this document. Look at the values in the second column and especially the notes you made to yourself in the third column. Develop an expression that gives the resale value of your car if you sell it after x years. Let's call it R(x). R (x ) = 200 00 ( 1 - ,15) 4 = 20, 000 ( ,85 )
