A common inhabitant of human intestines is the bacterium Escherichia coli, named after the German pediatrician Theodor Escherich, who identified it in 1885. A cell of this bacterium in a nutrient-broth medium divides into two cells every 20 minutes. The initial population of a culture is 40 cells. (a) Find the relative growth rate. k = 3ln (2) Correct: Your answer is correct. hr-1 (b) f cells after 8 hours. cells (d) Find the rate of growth after 8 hours. (Round your answer to the nearest integer.) cells/h (e) When will the population reach a million cells? h

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Mathematics

Kamryn Christensen

Today, 16:37

A common inhabitant of human intestines is the bacterium Escherichia coli, named after the German pediatrician Theodor Escherich, who identified it in 1885. A cell of this bacterium in a nutrient-broth medium divides into two cells every 20 minutes. The initial population of a culture is 40 cells. (a) Find the relative growth rate. k = 3ln (2) Correct: Your answer is correct. hr-1 (b) f cells after 8 hours. cells (d) Find the rate of growth after 8 hours. (Round your answer to the nearest integer.) cells/h (e) When will the population reach a million cells? h

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Zachary Young · Answer 1

A.) P (t) = P0exp (kt)

P (20/60) = 40 exp (20k/60)

80 = 40 exp (k/3)

exp (k/3) = 80/40 = 2

k/3 = ln (2)

k = 3ln (2)

b.) P (8) = 40 (2) ^24 = 40 (16777216) = 671088640 cells

d.) Rate of change = exp (8k) = exp (8 (3ln (2)) = exp (24ln (2)) = exp (16.6355) = 16777216 cells / hour

e.) P (t) = 40 (2) ^3t; t in hours

1,000,000 = 40 (8) ^t

25,000 = 8^t

ln (25,000) = t ln (8)

t = ln (25,000) / ln (8) = 4.87 hours