A common inhabitant of human intestines is the bacterium Escherichia coli. A cel

Question

A common inhabitant of human intestines is the bacterium Escherichia coli. A cell of this bacterium in a nutrient-broth medium divides into two cells every 20 minutes. The initial population of a culture is 54 cells. (

a) Find the relative growth rate. (Assume t is measured in hours.) k =

(b) Find an expression for the number of cells after t hours. P(t) =

(c) Find the number of cells after 8 hours. cells (d) Find the rate of growth after 8 hours. (Round your answer to three decimal places.) billion cells per hour (e) When will the population reach 20,000 cells? (Round your answer to two decimal places.) hr

Explanation / Answer

(a)

relative growth rate can be found from:
(e^k) - 1 = [N(t + 1) - N(t)]/N(t) and use t = 0
N(0) = 1
after 1 hr, N(1) = 8 (cells divide every 20 minutes)
e^k - 1 = [8 - 1]/1 = 7
e^k = 8
k = 2.997

b)

Using the relative growth rate we can write:
P(t) = 54*e^(2.997*t)

c) P(8) = 1.396*1012

d) rate of growth after 8hrs = [P(8) - P(0)]/P(0) = (1.396*1012 - 54) /54 = 25.851 billion cells/hr

e) 20000 = 54*e^(2.997*t)

5.9145 = 2.997*t

t = 1.973 hr

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