To find the population of a bacterial culture after a certain number of days given a specific growth rate, we can use the formula for exponential growth:
[tex]\[ y = P \times (1 + r)^t \][/tex]
Where:
- [tex]\( P \)[/tex] is the initial population,
- [tex]\( r \)[/tex] is the growth rate (as a decimal),
- [tex]\( t \)[/tex] is the time period in days,
- [tex]\( y \)[/tex] is the population after [tex]\( t \)[/tex] days.
Given:
- Initial population ([tex]\( P \)[/tex]) = 1,000 bacteria,
- Growth rate ([tex]\( r \)[/tex]) = 15% or 0.15 (as a decimal),
- Time period ([tex]\( t \)[/tex]) = 6 days.
Plugging these values into the formula, we get:
[tex]\[ y = 1,000 \times (1 + 0.15)^6 \][/tex]
[tex]\[ y = 1,000 \times (1.15)^6 \][/tex]
Calculating [tex]\( (1.15)^6 \)[/tex] results in approximately 2.313060765624999. Therefore:
[tex]\[ y = 1,000 \times 2.313060765624999 \approx 2,313.06 \][/tex]
The population of the culture 6 days after inoculation is approximately 2,313 bacteria.
The correct option from the provided choices is:
[tex]\[ y=1,000(1.15)^6 ; 2,313 \text{ bacteria} \][/tex]
