The number of bacteria $P(t)$ in a certain population increases according to the following function, where time $t$ is measured in hours.
\[
P(t)=1700 e^{0.25 t}
\]
Find the number of bacteria in the population after 4 hours and after 8 hours.
Round your answers to the nearest whole number as necessary.
Number after 4 hours:
bacteria
Number after 8 hours:
bacteria
Answer
Final Answer: The number of bacteria in the population after 4 hours is approximately \(\boxed{4621}\) and after 8 hours is approximately \(\boxed{12561}\).
Step 1 ：The problem is asking for the number of bacteria after 4 hours and 8 hours. This can be found by substituting \(t=4\) and \(t=8\) into the given function \(P(t)=1700 e^{0.25 t}\) and calculating the result.
Step 2 ：After substituting \(t=4\) into the function, we get \(P_4 = 1700 e^{0.25 \times 4} = 4621\) bacteria.
Step 3 ：After substituting \(t=8\) into the function, we get \(P_8 = 1700 e^{0.25 \times 8} = 12561\) bacteria.
Step 4 ：Final Answer: The number of bacteria in the population after 4 hours is approximately \(\boxed{4621}\) and after 8 hours is approximately \(\boxed{12561}\).
