1. **State the problem:** We need to model the temperature changes throughout the day starting at 6 a.m. with 50°F and graph the temperature based on the given rates of change. 2. **Identify time intervals and rates:** - From 6 a.m. to 8 a.m. (2 hours), temperature rises 4°F per hour. - From 8 a.m. to 11 a.m. (3 hours), temperature rises 2°F per hour. - From 11 a.m. to 6 p.m. (7 hours), temperature stays steady. - From 6 p.m. to 8 p.m. (2 hours), temperature drops 1°F per hour. - From 8 p.m. to midnight (4 hours), temperature drops steadily to 59°F. 3. **Calculate temperatures at key times:** - At 6 a.m.: $T=50$ - At 8 a.m.: $T=50 + 4 \times 2 = 50 + 8 = 58$ - At 11 a.m.: $T=58 + 2 \times 3 = 58 + 6 = 64$ - From 11 a.m. to 6 p.m.: temperature steady at $64$ - At 6 p.m.: $T=64$ - At 8 p.m.: $T=64 - 1 \times 2 = 64 - 2 = 62$ - From 8 p.m. to midnight (4 hours), temperature drops from 62 to 59, a total drop of 3°F over 4 hours. 4. **Find rate of temperature drop from 8 p.m. to midnight:** $$\text{Rate} = \frac{59 - 62}{4} = \frac{-3}{4} = -0.75 \text{°F per hour}$$ 5. **Write piecewise function for temperature $T(t)$ where $t$ is hours after 6 a.m.:** $$ T(t) = \begin{cases} 50 + 4t & 0 \leq t < 2 \\ 58 + 2(t-2) & 2 \leq t < 5 \\ 64 & 5 \leq t < 12 \\ 64 - (t-12) & 12 \leq t < 14 \\ 62 - 0.75(t-14) & 14 \leq t \leq 18 \end{cases} $$ 6. **Explanation:** - For $0 \leq t < 2$, temperature rises 4°F per hour from 50°F. - For $2 \leq t < 5$, temperature rises 2°F per hour starting at 58°F. - For $5 \leq t < 12$, temperature stays constant at 64°F. - For $12 \leq t < 14$, temperature drops 1°F per hour from 64°F. - For $14 \leq t \leq 18$, temperature drops 0.75°F per hour from 62°F to 59°F. This function models Jack's temperature data throughout the day. **Final answer:** The piecewise function above describes the temperature changes from 6 a.m. to midnight.