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What was the initial size of the culture? Find the doubling period. Find the population after 60 minutes. When will the population reach 12000.

What was the initial size of the culture? Find the doubling period. Find the population-example-1
User Juan Carlos Farah
by
3.2k points

1 Answer

14 votes
14 votes

The formula for exponential growth is expressed as

y = yo e^kt

where

yo is the initial population

y is the final population after time, t

t is the duration

when y = 500, t = 15

500 = yo e^15k

when y = 500, t = 15

1700 = yo e^40k

Dividing equation 2 by equation 1, we have

1700/500 = e^40k/e^15k

3.4 = e^(40k - 15k) = e^25k

Taking natural log of both sides

ln 3.4 = ln e^25k

25k = ln 3.4

k = ln 3.4/25

k = 0.049

The equation would be

y = yo e^0.049t

500 = yo e^15 *0.049

500 = yo e^0.735

yo = 500/e0.735

yo = 239.75

Doubling time is when y = 2yo Thus, we have

2yo = yo e^0.049t

2 * 239.75 = 239.75e^0.049t

479.5/239.75 = e^0.049t

2 = e^0.049t

Taking natural log of both sides

ln 2 = lne^0.049t

0.049t = ln2

t = ln2/0.049

t = 14.16

Doubling time = 14.16 minutes

For 60 minutes, t = 60

y = 239.75e^0.049*60

y = 4535.07

For y = 12000, we have

12000 = 239.75e^0.049t

12000/239.75 = e^0.049t

50.05 = e^0.049t

Taking natural log of both sides

ln 50.055 = lne^0.049t

0.049t = ln50.05

t = ln50.05/0.049

t = 79.86 minutes

User Mohit Bhansali
by
3.3k points