1. **State the problem:** We want to find how much an auto manufacturer should be willing to spend to keep a customer satisfied, given the customer will buy 10 more cars at $15000 each over 30 years, with a 20% net profit margin and a 9% discount rate. 2. **Formula and explanation:** The manufacturer’s expected profit from future sales is the present value (PV) of the profit from each car purchase. The profit per car is $15000 \times 0.20 = 3000$. Since the customer buys one car every 3 years for 30 years, there are 10 purchases. The present value of these profits is calculated using the formula for the present value of an annuity: $$PV = P \times \frac{1 - (1 + r)^{-n}}{r}$$ where: - $P = 3000$ (profit per car), - $r = 0.09$ (discount rate), - $n = 10$ (number of purchases). 3. **Calculate the present value:** Calculate $(1 + r)^{-n} = (1.09)^{-10}$. $$ (1.09)^{10} \approx 2.3674 \Rightarrow (1.09)^{-10} = \frac{1}{2.3674} \approx 0.4224 $$ Now calculate the fraction: $$ \frac{1 - 0.4224}{0.09} = \frac{0.5776}{0.09} \approx 6.418 $$ Multiply by $P$: $$ PV = 3000 \times 6.418 = 19254 $$ 4. **Interpretation:** The manufacturer can expect a present value profit of approximately 19254 from this customer over 30 years. Therefore, the manufacturer should be willing to spend up to $19254 to keep the customer satisfied. **Final answer:** $$\boxed{19254}$$