Step 1 :The function given is \(f(m)=500\left(\frac{1}{2}\right)^{\frac{m}{20}}\), which models the amount of a particular medicine in milligrams in a patient's bloodstream, $m$ minutes after the medicine is fully absorbed.
Step 2 :We need to find the time it takes for the amount of medicine to reduce by half. This means we need to find the value of $m$ when $f(m) = 250$ (since the initial amount is 500mg).
Step 3 :We set up the equation $250 = 500\left(\frac{1}{2}\right)^{\frac{m}{20}}$ and solve for $m$.
Step 4 :The solution to the equation is $m = 20$. This means it takes 20 minutes for the amount of medicine in the patient's bloodstream to reduce by half.
Step 5 :Final Answer: \(\boxed{20}\)