If $t$ is in minutes since the drug was administered, the concentration, $C(t)$ in $\mathrm{ng} / \mathrm{ml}$, of a drug in a patient's bloodstream is given by $C(t)=20 t e^{-0.05 t}$. (a) How long does it take for the drug to reach peak concentration? What is the peak concentration? Round your answers to one decimal place. The drug reaches its peak concentration at minutes. The peak concentration is $\mathrm{ng} / \mathrm{ml}$. (b) What is the concentration of the drug in the body after 15 minutes? After an hour? Round your answers to one decimal place. The concentration after 15 minutes is $\mathrm{ng} / \mathrm{ml}$. The concentration after 60 minutes is $\mathrm{ng} / \mathrm{ml}$ (c) If the minimum effective concentration is $10 \mathrm{ng} / \mathrm{ml}$, when should the next dose be administered? Enter your answer to the nearest 10 minutes. SUPPORT The next dose should be administered in minutes.

Step 1 ：Find the derivative of \(C(t) = 20t e^{-0.05t}\) using the product rule and the chain rule, which gives \(C'(t) = 20 e^{-0.05t} - t e^{-0.05t}\).

Step 2 ：Set \(C'(t)\) equal to zero and solve for \(t\), which gives \(20 e^{-0.05t} = t e^{-0.05t}\) and \(t = 20\) minutes.

Step 3 ：Substitute \(t = 20\) into \(C(t)\) to find the peak concentration, which gives \(C(20) = 20*20 e^{-0.05*20} = 326.9 \mathrm{ng/ml}\). So, \(\boxed{326.9 \mathrm{ng/ml}}\) is the peak concentration.

Step 4 ：Substitute \(t = 15\) and \(t = 60\) into \(C(t)\) to find the concentration of the drug after 15 minutes and 60 minutes, which gives \(C(15) = 20*15 e^{-0.05*15} = 247.6 \mathrm{ng/ml}\) and \(C(60) = 20*60 e^{-0.05*60} = 43.3 \mathrm{ng/ml}\). So, the concentration after 15 minutes is \(\boxed{247.6 \mathrm{ng/ml}}\) and the concentration after 60 minutes is \(\boxed{43.3 \mathrm{ng/ml}}\).

Step 5 ：Set \(C(t) = 10\) and solve for \(t\) to find when the next dose should be administered, which gives \(10 = 20t e^{-0.05t}\). Using a numerical method (like the Newton-Raphson method) gives \(t \approx 89\) minutes. So, the next dose should be administered in approximately \(\boxed{90}\) minutes (rounded to the nearest 10 minutes).