To determine the number of bacteria after 6 hours given the exponential growth function [tex]\( B(t) = 4 \cdot e^{0.8 t} \)[/tex], follow these steps:
1. Understand the Given Function:
The function used to model the bacteria growth is [tex]\( B(t) = 4 \cdot e^{0.8 t} \)[/tex]. Here:
- [tex]\( B(t) \)[/tex] is the number of bacteria at time [tex]\( t \)[/tex]
- [tex]\( 4 \)[/tex] is the initial number of bacteria (since at [tex]\( t = 0 \)[/tex], [tex]\( B(0) = 4 \)[/tex])
- [tex]\( e \)[/tex] is the base of the natural logarithm, approximately equal to 2.71828
- [tex]\( 0.8 \)[/tex] is the growth rate
2. Substitute the Given Time:
We need to find the number of bacteria at [tex]\( t = 6 \)[/tex] hours. Substitute [tex]\( t = 6 \)[/tex] into the function:
[tex]\[
B(6) = 4 \cdot e^{0.8 \cdot 6}
\][/tex]
Simplify the exponent:
[tex]\[
0.8 \cdot 6 = 4.8
\][/tex]
So the function becomes:
[tex]\[
B(6) = 4 \cdot e^{4.8}
\][/tex]
3. Calculate [tex]\( e^{4.8} \)[/tex]:
Using a calculator or any other method, find the approximate value of [tex]\( e^{4.8} \)[/tex]. The value of [tex]\( e^{4.8} \)[/tex] is approximately 121.5104.
4. Multiply by the Initial Amount:
Next, multiply the result by 4 (the initial number of bacteria):
[tex]\[
4 \cdot 121.5104 \approx 486.0416
\][/tex]
5. Round to the Nearest Integer:
Finally, round the result to the nearest integer:
[tex]\[
486.0416 \approx 486
\][/tex]
Thus, the approximate number of bacteria after 6 hours is [tex]\( \boxed{486} \)[/tex].
Therefore, the correct answer is:
B. 486
