Suppose a metal block is cooling so that its temperature $T$ (in ${ }^{\circ} \mathrm{C}$ ) is given by $T=200 \cdot 4^{-0.1 \mathrm{t}}+20$, 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 ${ }^{\circ} \mathrm{C}$.
(Simplify your answer. Do not round until the final answer. Then round to the nearest tenth as needed.)
Answer
\(\boxed{171.6}\) is the temperature of the metal block after 2 hours.
Step 1 ：Given the temperature formula \(T=200 \cdot 4^{-0.1t}+20\), where \(t\) is the time in hours.
Step 2 ：We are asked to find the temperature after 2 hours, so we substitute \(t=2\) into the formula.
Step 3 ：This gives us \(T=200 \cdot 4^{-0.1 \cdot 2}+20\).
Step 4 ：Solving this expression gives us \(T=171.5716566510398\).
Step 5 ：Rounding to the nearest tenth, we get \(T=171.6\).
Step 6 ：\(\boxed{171.6}\) is the temperature of the metal block after 2 hours.
