To answer the question of what the approximate number of bacteria will be after 6 hours given the function [tex]\( B(t) = 4 \cdot e^{0.8t} \)[/tex], let's break it down step-by-step.
1. Understand the exponential growth function:
The function [tex]\( B(t) = 4 \cdot e^{0.8t} \)[/tex] describes how the bacteria population grows over time. Here,
- [tex]\( B(t) \)[/tex] is the number of bacteria at time [tex]\( t \)[/tex].
- 4 is the initial number of bacteria when [tex]\( t = 0 \)[/tex].
- [tex]\( e \)[/tex] is the base of the natural logarithm.
- [tex]\( 0.8 \)[/tex] is the growth rate per hour.
2. Substitute the time [tex]\( t = 6 \)[/tex] into the function:
We need to find [tex]\( B(6) \)[/tex]. So,
[tex]\[
B(6) = 4 \cdot e^{0.8 \cdot 6}
\][/tex]
3. Calculate the exponent:
First, calculate the exponent:
[tex]\[
0.8 \cdot 6 = 4.8
\][/tex]
4. Exponentiate [tex]\( e \)[/tex] to the power of 4.8:
Find [tex]\( e^{4.8} \)[/tex]. Using a calculator, you would get:
[tex]\[
e^{4.8} \approx 121.51042
\][/tex]
5. Multiply the initial number of bacteria by this value:
Now multiply this result by 4, the initial number of bacteria:
[tex]\[
4 \cdot 121.51042 = 486.04168
\][/tex]
6. Round the result to the nearest whole number:
The approximate number of bacteria after 6 hours is:
[tex]\[
486
\][/tex]
Therefore, the correct answer is A. 486.
