Morphine is a drug that is widely used for pain management. However, morphine can cause fatal respiratory arrest. Since pain perception and drug tolerance vary with patients, morphine is gradually administered with small increments until pain is controlled or side effects begin to appear.

In one intravenous infusion, morphine was injected continuously at an increasing rate of t milligrams per hour. Suppose that the body removes the drug at a rate proportional to the amount of the drug present in the body, with constant of proportionality k = .35. Let f (t) denote the amount of morphine in the body, t hours from the beginning of the infusion.

(a) Find a differential equation satisfied by f (t).

(b) Assuming that the infusion lasted 8 hours, determine the amount of morphine in the body during the infusion if at the beginning of the infusion the body was free of morphine.