Question

1. a biologist puts an initial population of 500 bacteria into a growth plate. the population is expected to double every 4 hours. which of the following equations gives the expected number of bacteria, n, after x days? (24 hours = 1 day)

a. ( n = 500(2)^{x} )
b. ( n = 500(2)^{6x} )
c. ( n = 500(6)^{x} )
d. ( n = 500(6)^{2x} )

Explanation:

Step1: Calculate the number of 4 - hour intervals in a day

Since there are 24 hours in a day and the population doubles every 4 hours, the number of 4 - hour intervals in a day is $\frac{24}{4}=6$.

Step2: Analyze the exponential growth formula

The general formula for exponential growth is $n = n_0\cdot a^{kt}$, where $n_0$ is the initial population, $a$ is the base of the exponential (in case of doubling, $a = 2$), $k$ is the growth factor per unit time, and $t$ is the time. Here, $n_0=500$, and the growth factor per day (since the time unit $t$ is in days) is $k = 6$ (because of 6 four - hour intervals in a day). So the formula for the number of bacteria $n$ after $x$ days is $n=500\times(2)^{6x}$.

Answer:

B. $n = 500(2)^{6x}$