If a substance is injected into the bloodstream, the percent of the maximum dosage that is present at time $t$ is given by $y=100\left(1-e^{-0.37(16-t)}\right)$, where $t$ is in hours, with $0 \leq t \leq 16$. In how many hours will the percent reach $67 \%$ ?
The percent will reach $67 \%$ in $\square$ hours.
Answer
Final Answer: The percent will reach \(67\%\) in approximately \(\boxed{13}\) hours.
Step 1 ：We are given the equation \(y=100\left(1-e^{-0.37(16-t)}\right)\), where \(y\) is the percent of the maximum dosage that is present at time \(t\) (in hours), and \(0 \leq t \leq 16\). We want to find the time \(t\) when the percent \(y\) reaches \(67\%\).
Step 2 ：We can set up the equation as \(100\left(1-e^{-0.37(16-t)}\right) = 67\).
Step 3 ：Solving this equation symbolically, we get multiple solutions, many of which are complex numbers. However, in this context, we are only interested in the real solution.
Step 4 ：The real solution is approximately \(t = 13.00361452832\) hours.
Step 5 ：However, since the time \(t\) is usually measured in whole hours, we can round this to the nearest whole number.
Step 6 ：Final Answer: The percent will reach \(67\%\) in approximately \(\boxed{13}\) hours.
