The amount of medicine (in milligrams) in a patient's bloodstream $t$ hours since taking a pill can be modeled by $y=390(0.85)^{t}$. The patient must take another pill when the amount of medicine in her bloodstream reaches 150 milligrams. How long since taking the first pill does the patient need to take the second pill? Round only your final answer to the second decimal place or enter your answer without using a calculator. Preview

Step 1 ：We are given the equation \(y=390(0.85)^{t}\) which models the amount of medicine in a patient's bloodstream t hours after taking a pill.

Step 2 ：We need to find the time t when the amount of medicine y in the patient's bloodstream reaches 150 milligrams.

Step 3 ：This can be done by setting \(y=150\) in the given equation and solving for \(t\).

Step 4 ：Substituting \(y=150\) into the equation, we get \(150=390(0.85)^{t}\).

Step 5 ：Solving this equation for \(t\), we get \(t=5.88\).

Step 6 ：Final Answer: The patient needs to take the second pill approximately \(\boxed{5.88}\) hours after taking the first pill.