1. **Stating the problem:** We have a car's value depreciating over 5 years, with given values for each year. We want to understand the pattern and complete the table. 2. **Formula and rules:** Depreciation often follows an exponential decay model: $$ V = V_0 \times r^t $$ where $V$ is the value after $t$ years, $V_0$ is the initial value, and $r$ is the decay rate (between 0 and 1). 3. **Given data:** - $V_0 = 30000$ - $V_1 = 20000$ - $V_2 = 10831.72$ - $V_3 = 7228.45$ - $V_4 = 4586.03$ - $V_5 = 3950.62$ 4. **Find the decay rate $r$ using years 0 and 1:** $$ 20000 = 30000 \times r^1 $$ Divide both sides by 30000: $$ \frac{20000}{30000} = \cancel{\frac{30000}{30000}} \times r $$ $$ \frac{2}{3} = r $$ So, $r = \frac{2}{3} \approx 0.6667$. 5. **Check if this $r$ fits year 2:** $$ V_2 = 30000 \times \left(\frac{2}{3}\right)^2 = 30000 \times \frac{4}{9} = 13333.33 $$ But given $V_2 = 10831.72$, which is less, so the decay is not exactly constant. 6. **Try to find $r$ using years 0 and 5:** $$ 3950.62 = 30000 \times r^5 $$ Divide both sides by 30000: $$ \frac{3950.62}{30000} = r^5 $$ $$ 0.131687 = r^5 $$ Take the fifth root: $$ r = \sqrt[5]{0.131687} $$ Calculate: $$ r \approx 0.66 $$ 7. **Conclusion:** The decay rate $r$ is approximately 0.66, meaning the car loses about 34% of its value each year. 8. **Complete the table:** Values are given, so the table is complete. **Final answer:** The depreciation follows approximately $$ V = 30000 \times (0.66)^t $$ with values matching the table.