To estimate when the population of bacteria will exceed 1199, follow these steps:
1. Understand the Given Function: The population of the bacteria is given by the function \(P(t) = 450 e^{0.2 t}\), where \( P(t) \) is the population at time \( t \).
2. Set Up the Inequality: We want to find the time \( t \) when the population exceeds 1199. This translates to solving the inequality:
[tex]\[
450 e^{0.2 t} > 1199
\][/tex]
3. Convert to Equation: First solve the related equation to find the exact time when the population reaches 1199:
[tex]\[
450 e^{0.2 t} = 1199
\][/tex]
4. Solve for \( t \):
- To isolate \( t \), divide both sides by 450:
[tex]\[
e^{0.2 t} = \frac{1199}{450}
\][/tex]
- Calculate the right-hand side:
[tex]\[
e^{0.2 t} \approx 2.664
\][/tex]
- Take the natural logarithm of both sides to solve for \( t \):
[tex]\[
0.2 t = \ln(2.664)
\][/tex]
- Calculate \( \ln(2.664) \) using a calculator:
[tex]\[
\ln(2.664) \approx 0.9808
\][/tex]
- Finally, solve for \( t \) by dividing by 0.2:
[tex]\[
t \approx \frac{0.9808}{0.2} = 4.904
\][/tex]
5. Round to One Decimal Place: Round the result to one decimal place to get the final answer:
[tex]\[
t \approx 4.9
\][/tex]
Therefore, the population of bacteria will exceed 1199 at approximately [tex]\( t = 4.9 \)[/tex] hours.
