The number of bacteria [tex]P(t)[/tex] in a certain population increases according to the following function, where time [tex]t[/tex] is measured in hours.

[tex]\[
P(t) = 2800 e^{0.06 t}
\][/tex]

Find the number of bacteria in the population after 4 hours and after 7 hours. Round your answers to the nearest whole number as necessary.

Number after 4 hours: [tex]\(\square\)[/tex] bacteria

Number after 7 hours: [tex]\(\square\)[/tex] bacteria



Answer :

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]