1. **Problem statement:** A freezer starts at $-14^\circ C$ and warms up by $3^\circ C$ per hour after being unplugged. When the temperature rises above $0^\circ C$, it is plugged back in and cools down by $4^\circ C$ per hour. Find the temperature 8 hours after unplugging. 2. **Warming phase:** The temperature after $t$ hours unplugged is given by: $$ T(t) = -14 + 3t $$ We want to find when $T(t)$ first exceeds $0$: $$ -14 + 3t > 0 $$ $$ 3t > 14 $$ $$ t > \frac{14}{3} \approx 4.67 \text{ hours} $$ So after 5 hours (since checks are hourly), the temperature is above $0^\circ C$ and the freezer is plugged back in. 3. **Calculate temperature at 5 hours:** $$ T(5) = -14 + 3 \times 5 = -14 + 15 = 1^\circ C $$ 4. **Cooling phase:** After plugging back in at hour 5, the freezer cools by $4^\circ C$ per hour. Let $t_c$ be hours after plugging back in. For $t_c = 0,1,2,3$ (hours 5 to 8): $$ T_{cool}(t_c) = 1 - 4t_c $$ 5. **Calculate temperature at 8 hours (3 hours after plugging back in):** $$ T_{cool}(3) = 1 - 4 \times 3 = 1 - 12 = -11^\circ C $$ **Final answer:** The temperature 8 hours after unplugging is $\boxed{-11^\circ C}$. --- **Summary table:** | Time (hours) | Temperature ($^\circ C$) | |--------------|--------------------------| | 0 | -14 | | 1 | -11 | | 2 | -8 | | 3 | -5 | | 4 | -2 | | 5 | 1 | | 6 | -3 | | 7 | -7 | | 8 | -11 |