1. **Problem statement:** Find the average velocity over given intervals and the instantaneous velocity at $t=8$ for the displacement function $$s(t) = \frac{1}{2}t^2 - 5t + 17$$ where $t$ is in seconds and $s$ in meters. 2. **Formula for average velocity:** Average velocity over $[t_1, t_2]$ is $$v_{avg} = \frac{s(t_2) - s(t_1)}{t_2 - t_1}$$ 3. **Calculate $s(t)$ values:** $$s(t) = \frac{1}{2}t^2 - 5t + 17$$ 4. **Calculate average velocities:** (i) For $[4,8]$: $$s(4) = \frac{1}{2}(4)^2 - 5(4) + 17 = 8 - 20 + 17 = 5$$ $$s(8) = \frac{1}{2}(8)^2 - 5(8) + 17 = 32 - 40 + 17 = 9$$ $$v_{avg} = \frac{9 - 5}{8 - 4} = \frac{4}{4} = 1\ \text{m/s}$$ (ii) For $[6,8]$: $$s(6) = \frac{1}{2}(6)^2 - 5(6) + 17 = 18 - 30 + 17 = 5$$ $$v_{avg} = \frac{9 - 5}{8 - 6} = \frac{4}{2} = 2\ \text{m/s}$$ (iii) For $[8,10]$: $$s(10) = \frac{1}{2}(10)^2 - 5(10) + 17 = 50 - 50 + 17 = 17$$ $$v_{avg} = \frac{17 - 9}{10 - 8} = \frac{8}{2} = 4\ \text{m/s}$$ (iv) For $[8,12]$: $$s(12) = \frac{1}{2}(12)^2 - 5(12) + 17 = 72 - 60 + 17 = 29$$ $$v_{avg} = \frac{29 - 9}{12 - 8} = \frac{20}{4} = 5\ \text{m/s}$$ 5. **Formula for instantaneous velocity:** Instantaneous velocity is the derivative of $s(t)$: $$v(t) = s'(t) = \frac{d}{dt}\left( \frac{1}{2}t^2 - 5t + 17 \right) = t - 5$$ 6. **Calculate instantaneous velocity at $t=8$:** $$v(8) = 8 - 5 = 3\ \text{m/s}$$ **Final answers:** - Average velocities: (i) 1 m/s (ii) 2 m/s (iii) 4 m/s (iv) 5 m/s - Instantaneous velocity at $t=8$: 3 m/s