Suppose a metal block is cooling so that its temperature $T$ (in ${ }^{\circ} \mathrm{C}$ ) is given by $T=200 \cdot 4^{-0.1 t}+20$, where $t$ is in hours.
a. Find the temperature after (i) 2 hours, (ii) 3.5 hours.
b. How long has the cooling been taking place if the block now has a temperature of $120^{\circ} \mathrm{C}$ ?
c. Find the eventual temperature $(t \rightarrow \infty)$.
a. (i) After 2 hours the temperature will be about $171.6^{\circ} \mathrm{C}$.
(Simplify your answer. Do not round until the final answer. Then round to the nearest tenth as needed.)
(ii) After 3.5 hours the temperature will be about $\square^{\circ} \mathrm{C}$.
(Simplify your answer. Do not round until the final answer. Then round to the nearest tenth as needed.)
Answer
Final Answer: After 3.5 hours the temperature will be about \(\boxed{143.1^{\circ} \mathrm{C}}\).
Step 1 ：To find the temperature after 3.5 hours, we need to substitute \(t=3.5\) into the given equation \(T=200 \cdot 4^{-0.1 t}+20\).
Step 2 ：Substituting \(t=3.5\) into the equation, we get \(T = 200 \cdot 4^{-0.1 \cdot 3.5}+20\).
Step 3 ：Solving the equation, we get \(T = 143.11444133449163\).
Step 4 ：Rounding to the nearest tenth, we get \(T = 143.1\).
Step 5 ：Final Answer: After 3.5 hours the temperature will be about \(\boxed{143.1^{\circ} \mathrm{C}}\).
