1. **State the problem:** We are given data points for the value of a car (in thousands of dollars) at different ages (in years) and need to write a linear function $C(t)$ that models the car's value based on its age $t$. 2. **Identify points:** From the data, two points are $(t_1, C_1) = (5, 13.5)$ and $(t_2, C_2) = (11.4, 5)$. 3. **Find the slope $m$ of the line:** $$m = \frac{C_2 - C_1}{t_2 - t_1} = \frac{5 - 13.5}{11.4 - 5} = \frac{-8.5}{6.4} = -1.328125$$ 4. **Write the linear equation in point-slope form:** $$C - C_1 = m(t - t_1)$$ $$C - 13.5 = -1.328125(t - 5)$$ 5. **Simplify to slope-intercept form:** $$C = -1.328125t + 1.328125 \times 5 + 13.5$$ $$C = -1.328125t + 6.640625 + 13.5$$ $$C = -1.328125t + 20.140625$$ 6. **Interpret $C(6)$:** $C(6)$ is the value of the car when it is 6 years old. 7. **Calculate $C(6)$:** $$C(6) = -1.328125 \times 6 + 20.140625 = -7.96875 + 20.140625 = 12.171875$$ **Final answers:** $$C(t) = -1.328125t + 20.140625$$ $$C(6) = 12.171875$$ This means the car is worth approximately 12.17 thousand dollars at 6 years old.