Question
If the number of bacteria in a colony doubles every 384 hours and there is currently a population of 2,150 bacteria, what will the population be 1,152 hours from now?
bacteria
bacteria
Answers
GPT-4o mini
To find the population of bacteria after 1,152 hours, we first need to determine how many times the population will double in that time period. Since the population doubles every 384 hours, we can calculate the number of doublings that occur in 1,152 hours using the formula:
\[
\text{Number of doublings} = \frac{\text{Total time}}{\text{Time per doubling}} = \frac{1152 \, \text{hours}}{384 \, \text{hours}} = 3
\]
This means the population will double 3 times in 1,152 hours.
Now, we can calculate the population after 3 doublings. The population doubles every time, so we use the formula for exponential growth:
\[
\text{New population} = \text{Initial population} \times 2^{\text{Number of doublings}}
\]
Substituting the values we have:
\[
\text{New population} = 2150 \times 2^{3}
\]
Calculating \(2^3\):
\[
2^3 = 8
\]
Now, multiplying by the initial population:
\[
\text{New population} = 2150 \times 8 = 17200
\]
Therefore, the population of the bacteria after 1,152 hours will be:
\[
\boxed{17200}
\]
\[
\text{Number of doublings} = \frac{\text{Total time}}{\text{Time per doubling}} = \frac{1152 \, \text{hours}}{384 \, \text{hours}} = 3
\]
This means the population will double 3 times in 1,152 hours.
Now, we can calculate the population after 3 doublings. The population doubles every time, so we use the formula for exponential growth:
\[
\text{New population} = \text{Initial population} \times 2^{\text{Number of doublings}}
\]
Substituting the values we have:
\[
\text{New population} = 2150 \times 2^{3}
\]
Calculating \(2^3\):
\[
2^3 = 8
\]
Now, multiplying by the initial population:
\[
\text{New population} = 2150 \times 8 = 17200
\]
Therefore, the population of the bacteria after 1,152 hours will be:
\[
\boxed{17200}
\]