1. **State the problem:** We have a piecewise function for temperature $T$ in °C as a function of time $t$ in seconds: $$f(t) = \begin{cases} 40 + 2t & 0 \leq t \leq 30 \\ 130 - t & 30 \leq t \leq 60 \end{cases}$$ We need to find: - The initial temperature (at $t=0$). - The temperature after 60 seconds. - The maximum temperature reached. - Sketch the graph. - The time interval when temperature $\geq 80$°C. 2. **Find the initial temperature:** At $t=0$, use the first piece: $$T = 40 + 2 \times 0 = 40$$ So, the initial temperature is **40°C**. 3. **Find the temperature after 60 seconds:** At $t=60$, use the second piece: $$T = 130 - 60 = 70$$ So, the temperature after 60 seconds is **70°C**. 4. **Find the maximum temperature:** The function increases from $t=0$ to $t=30$ and then decreases from $t=30$ to $t=60$. At $t=30$, check temperature: $$T = 40 + 2 \times 30 = 40 + 60 = 100$$ At $t=30$, the temperature is 100°C, which is the peak. So, the maximum temperature is **100°C**. 5. **Sketch the graph:** - From $(0,40)$ to $(30,100)$, the graph is a straight line increasing. - From $(30,100)$ to $(60,70)$, the graph is a straight line decreasing. 6. **Find the time interval when $T \geq 80$°C:** - For $0 \leq t \leq 30$, solve: $$40 + 2t \geq 80$$ $$2t \geq 40$$ $$t \geq 20$$ So, on this interval, $t \in [20,30]$. - For $30 \leq t \leq 60$, solve: $$130 - t \geq 80$$ $$-t \geq -50$$ $$t \leq 50$$ So, on this interval, $t \in [30,50]$. Combining both intervals: $$t \in [20,50]$$ **Final answers:** - Initial temperature: **40°C** - Temperature after 60 seconds: **70°C** - Maximum temperature: **100°C** - Time interval with $T \geq 80$°C: **20 seconds to 50 seconds**