Question: When a patient with hypertension takes a particular type of blood pressure medication, the effects on the systolic pressure, $S(t)$, can be measured by the following piecewise defined function: $S(t) = \begin{cases} 140 - 6t & 0 \leq t \leq 4 \\ 116 & 4 < t < 8 \\ 96 + 8t & 8 \leq t \leq 12 \end{cases}$ where $t$ is the time, in hours, since taking the medication. Based on the graph of the piecewise function, if the patient takes the blood pressure medication at 8 a.m., in which interval will their systolic pressure be lowest? 10 a.m. to 12 p.m. 12 p.m. to 2 p.m. 4 p.m. to 6 p.m. 8 p.m. to 10 p.m.

1. **State the problem:** We want to find the time interval after the patient takes the medication at 8 a.m. when the systolic pressure $S(t)$ is lowest. 2. **Understand the piecewise function:** $$ S(t) = \begin{cases} 140 - 6t & 0 \leq t \leq 4 \\ 116 & 4 < t < 8 \\ 96 + 8t & 8 \leq t \leq 12 \end{cases} $$ Here, $t$ is hours since taking the medication at 8 a.m. 3. **Analyze each interval:** - For $0 \leq t \leq 4$: $S(t) = 140 - 6t$ decreases from $140$ at $t=0$ to $140 - 6 \times 4 = 116$ at $t=4$. - For $4 < t < 8$: $S(t) = 116$ constant. - For $8 \leq t \leq 12$: $S(t) = 96 + 8t$ increases from $96 + 8 \times 8 = 160$ at $t=8$ to $96 + 8 \times 12 = 192$ at $t=12$. 4. **Find the lowest systolic pressure:** - Between $0$ and $4$ hours, $S(t)$ decreases to $116$. - Between $4$ and $8$ hours, $S(t)$ stays at $116$. - Between $8$ and $12$ hours, $S(t)$ increases from $160$ to $192$. The lowest value is $116$, occurring from $t=4$ to $t=8$ hours after medication. 5. **Convert $t$ to clock time:** - Medication taken at 8 a.m. - $t=4$ corresponds to 12 p.m. - $t=8$ corresponds to 4 p.m. 6. **Answer:** The systolic pressure is lowest between 12 p.m. and 4 p.m. **Given options:** - 10 a.m. to 12 p.m. - 12 p.m. to 2 p.m. - 4 p.m. to 6 p.m. - 8 p.m. to 10 p.m. The interval $12$ p.m. to $2$ p.m. is within the lowest pressure interval $12$ p.m. to $4$ p.m., so the correct choice is **12 p.m. to 2 p.m.**.