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.