Question 4.4.8.2: Bacterial Growth A colony of bacteria increases according to......
Bacterial Growth
A colony of bacteria increases according to the law of uninhibited growth.
(a) If N is the number of cells and t is the time in hours, express N as a function of t.
(b) If the number of bacteria doubles in 3 hours, find the function that gives the number of cells in the culture.
(c) How long will it take for the size of the colony to triple?
(d) How long will it take for the population to double a second time (that is, increase four times)?
(a) Using formula (2), the number N of cells at time t is
\mathbf{N}(t)=N_{\mathrm{0}}e^{k t} k>0 (2)
where N_{0} is the initial number of bacteria present and k is a positive number.
(b) To find the growth rate k. note that the number of cells doubles in 3 hours Hence
\mathbf{N}(3)=2N_{\mathrm{0}}But N(3)\,=\,N_{0}e^{k(3)}, so
N_{0}e^{k(3)}\ =\ 2N_{0}e^{k(3)} = 2 Divide both sides by N_{0}
3 k = \ln 2 Write the exponential equation as a logarithm.
k={\frac{1}{3}}\ln2\ \approx0.23105The function that models this growth process is therefore
N(t)\,=\,N_{0}e^{0.23105t}(c) The time t needed for the size of the colony to triple requires that N = 3N_{0} Substitute 3N_0 for N to get
3N_{0}=N_{0}e^{0.23105t}3=e^{0.23105t} Divide both sides by N_{0}
0.23105t=\ln3 Write the exponential equation as a logarithm.
t={\frac{\ln3}{0.23105}}\approx4.755\,\mathrm{hours}It will take about 4.755 hours or 4 hours, 45 minutes for the size of the colony to triple.
(d) If a population doubles in 3 hours, it will double a second time in 3 more hours, for a total time of 6 hours.