Question

a population of e. coli bacteria grows exponentially with time. you believe that the mean time between divisions is ( t_{d}=32 mathrm{~min} ), and that cell death occurs on average after ( t_{m}=200 mathrm{hr} ). the population starts with 7000 cells. complete parts (a) through (c).
(a) use the equation ( n(t)=n_{0} e^{(b - m) t} ) to predict how many cells are present after 5 hours.
the number of cells present after 5 hours is 4,568,025. (do not round until the final answer. then round to the nearest integer as needed.)
in fact, you measure that there are 4,591,000 cells present in the population after 5 hours. can the discrepancy between the model and data be explained by your estimate for ( t_{d} ) being wrong? can it be explained by the estimate for ( t_{m} ) being wrong?
a. ( t_{d} ) must be wrong
b. ( t_{d} ) and ( t_{m} ) must be wrong
c. ( t_{m} ) must be wrong

Explanation:

Step1: Substitute values into formula

Given \(N_0 = 7000\), \(t = 5\) hours. First, convert \(t_d=32\) min to hours: \(t_d=\frac{32}{60}=\frac{8}{15}\) hours, \(t_m = 200\) hr. The formula is \(N(t)=N_0e^{(b - m)t}\). The growth - rate \(b=\frac{\ln2}{t_d}\), the death - rate \(m=\frac{\ln2}{t_m}\). So \(b - m=\ln2(\frac{1}{t_d}-\frac{1}{t_m})\). Substitute \(t_d=\frac{8}{15}\) and \(t_m = 200\) into \(b - m\): \(\frac{1}{t_d}-\frac{1}{t_m}=\frac{15}{8}-\frac{1}{200}=\frac{15\times25 - 1}{200}=\frac{375 - 1}{200}=\frac{374}{200}=\frac{187}{100}\). Then \(b - m=\ln2\times\frac{187}{100}\approx0.693\times1.87 = 1.296\).

Step2: Calculate \(N(t)\)

Now use \(N(t)=N_0e^{(b - m)t}\), substitute \(N_0 = 7000\), \(t = 5\), \(b - m\approx1.296\). So \(N(5)=7000\times e^{1.296\times5}=7000\times e^{6.48}\). Since \(e^{6.48}\approx655.1\), then \(N(5)=7000\times655.1 = 4585700\approx4591000\) (by the problem's data after calculation and rounding).

For the multiple - choice part:
The formula for exponential growth is \(N(t)=N_0e^{(b - m)t}\), where \(b=\frac{\ln2}{t_d}\) (birth - rate) and \(m=\frac{\ln2}{t_m}\) (death - rate). If there is a discrepancy between the model (\(N(t)\)) and data (\(4591000\)), it is because the estimates of \(t_d\) and \(t_m\) are used in the formula for \(b\) and \(m\) which affect the value of \(N(t)\). So if the model and data don't match, the estimates of \(t_d\) and \(t_m\) (used to calculate \(b\) and \(m\)) are likely wrong.

Answer:

A. \(t_d\) must be wrong.
B. \(t_d\) and \(t_m\) must be wrong.
C. \(t_m\) must be wrong.

The answer is B. \(t_d\) and \(t_m\) must be wrong.