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 \(\boxed{143.1^{\circ} \mathrm{C}}\).