To determine the number of bacteria in the population given by the function [tex]\( P(t) = 2800 e^{0.06t} \)[/tex], we need to evaluate this function at specific time points, namely after 4 hours and after 7 hours.
Here are the steps to solve the problem:
1. Define the given function for the number of bacteria:
[tex]\[
P(t) = 2800 e^{0.06t}
\][/tex]
2. Calculate the number of bacteria after 4 hours:
- Substitute [tex]\( t = 4 \)[/tex] into the function:
[tex]\[
P(4) = 2800 \cdot e^{0.06 \cdot 4}
\][/tex]
- Calculate the exponent:
[tex]\[
0.06 \cdot 4 = 0.24
\][/tex]
- Now plug this value back into the exponential function and multiply by 2800:
[tex]\[
P(4) = 2800 \cdot e^{0.24}
\][/tex]
- Evaluating this, we get:
[tex]\[
P(4) \approx 3559
\][/tex]
3. Calculate the number of bacteria after 7 hours:
- Substitute [tex]\( t = 7 \)[/tex] into the function:
[tex]\[
P(7) = 2800 \cdot e^{0.06 \cdot 7}
\][/tex]
- Calculate the exponent:
[tex]\[
0.06 \cdot 7 = 0.42
\][/tex]
- Now plug this value back into the exponential function and multiply by 2800:
[tex]\[
P(7) = 2800 \cdot e^{0.42}
\][/tex]
- Evaluating this, we get:
[tex]\[
P(7) \approx 4261
\][/tex]
Thus, rounding to the nearest whole number, the number of bacteria after 4 hours is 3,559 and after 7 hours is 4,261.
The final answers are:
- Number of bacteria after 4 hours: [tex]\( 3,559 \)[/tex]
- Number of bacteria after 7 hours: [tex]\( 4,261 \)[/tex]
