Question

7 bacteria are being grown in a petri dish in a biology lab. the number of bacteria in the culture after a given number of hours is shown in the table below.
assuming this exponential trend continues, is it reasonable to expect at least 3500 bacteria at hour 7? justify your answer.

Explanation:

Step1: Calculate the growth factor

First, find the ratio between consecutive bacteria counts.
For hour 1 to 2: $\frac{2200}{1990}\approx1.1055$
For hour 2 to 3: $\frac{2430}{2200} = 1.1045$
For hour 3 to 4: $\frac{2685}{2430}=1.1049$
For hour 4 to 5: $\frac{2965}{2685}\approx1.1043$
The average growth factor $r\approx1.1048$.

Step2: Use the exponential formula

The general exponential formula is $N = N_0\times r^{t - t_0}$. Let $t_0 = 1$, $N_0=1990$.
For $t = 7$:
$N=1990\times(1.1048)^{7 - 1}$
$N=1990\times(1.1048)^{6}$
Calculate $(1.1048)^{6}\approx1.807$
Then $N = 1990\times1.807\approx3606$

Answer:

Yes, it is reasonable. Using the exponential growth model with an average growth factor of approximately $1.1048$, the number of bacteria at hour 7 is approximately $3606$, which is more than $3500$.