1. **Problem statement:** A car manufacturer sells a new car for 15000 and expects the customer to buy 10 more cars over 30 years, each at 15000. The net profit margin is 20%, and the discount rate is 9%. We want to find how much the manufacturer should spend to keep the customer satisfied. 2. **Formula and explanation:** The manufacturer’s expected profit is the present value of the profit from future car sales. Profit per car = $15000 \times 0.20 = 3000$ The customer buys 10 cars over 30 years, one every 3 years, so payments occur at years 3, 6, 9, ..., 30. We use the Present Value of an annuity formula for discrete payments: $$PV = P \times \frac{1 - (1 + r)^{-n}}{r}$$ where $P$ is the profit per period, $r$ is the discount rate per period, and $n$ is the number of periods. 3. **Calculate parameters:** - $P = 3000$ - $r = 0.09$ (annual discount rate) - $n = 10$ (number of cars) 4. **Calculate present value:** $$PV = 3000 \times \frac{1 - (1 + 0.09)^{-10}}{0.09}$$ Calculate $(1 + 0.09)^{-10}$: $$1.09^{-10} = \frac{1}{1.09^{10}} \approx \frac{1}{2.3674} \approx 0.4224$$ So, $$PV = 3000 \times \frac{1 - 0.4224}{0.09} = 3000 \times \frac{0.5776}{0.09} = 3000 \times 6.418$$ $$PV \approx 19254$$ 5. **Interpretation:** The manufacturer can expect a present value profit of about 19254 from this customer’s future purchases. 6. **Answer:** The manufacturer should be willing to spend up to about 19254 to keep the customer satisfied.