1. **Problem statement:** We are given a piecewise function for the car's speed $v(t)$ over the first 8 seconds: $$v(t) = \begin{cases} 8e^{0.4t} - 8, & 0 \leq t \leq 4 \\ -t^2 + 24t - 48.4, & 4 < t \leq 8 \end{cases}$$ We need to find the values of $v(t)$ at $t=5,6,7,8$ to complete the table. 2. **Recall:** For $t$ between 0 and 4, use the exponential formula. For $t$ between 4 and 8, use the quadratic formula. 3. **Calculate $v(5)$:** Since $5 > 4$, use the quadratic formula: $$v(5) = -(5)^2 + 24 \times 5 - 48.4 = -25 + 120 - 48.4 = 46.6$$ 4. **Calculate $v(6)$:** $$v(6) = -(6)^2 + 24 \times 6 - 48.4 = -36 + 144 - 48.4 = 59.6$$ 5. **Calculate $v(7)$:** $$v(7) = -(7)^2 + 24 \times 7 - 48.4 = -49 + 168 - 48.4 = 70.6$$ 6. **Calculate $v(8)$:** $$v(8) = -(8)^2 + 24 \times 8 - 48.4 = -64 + 192 - 48.4 = 79.6$$ 7. **Completed table:** | Time $t$ (seconds) | Speed $v(t)$ (km/hour) | |--------------------|------------------------| | 0 | 0 | | 1 | 9.8 | | 2 | 31.6 | | 3 | 70.6 | | 4 | 79.6 | | 5 | 46.6 | | 6 | 59.6 | | 7 | 70.6 | | 8 | 79.6 | 8. **Graphing:** The graph of $y = v(t)$ is piecewise with an exponential growth from $t=0$ to $t=4$ and a quadratic curve from $t=4$ to $t=8$ as given by the formulas. This completes the solution for the first question.