25/26 A bacteria culture grows with a constant relative growth rate. After 2 hou

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25/26

A bacteria culture grows with a constant relative growth rate. After 2 hours there are 600 bacteria and after 8 hours the count is 75,000. Find the initial population. Find an expression for the population after t hours. Find the number of cells after 3 hours. (Round your answer to the nearest integer.) Find the rate of growth after 3 hours. (Round your answer to the nearest integer.) When will the population reach 200,000? (Round your answer to one decimal place.) Use the fact that the world population was 2560 million in 1950 and 3040 million in 1960 to model the population of the world in the second half of the 20th century. (Assume that the growth rate is proportional to the population size.) What is the relative growth rate? Use the model to estimate the world population in 1995 and to predict the population in the year 2020. SOLUTION We measure the time t in years and let t = 0 in the year 1950. We measure the population P(t) in millions of people. Then P(0) = 2560 and P(10) = 3040. Since we are assuming that dP/dt = kP, this theorem gives the following. (Round to six decimal places.) The relative growth rate is about 1.7% per year and the model is P(t) = We estimate that the world population in 1995 was

Explanation / Answer

a) so we have

600 = A e^(2k)

and
75000= A e^(8k)

divide equations

75000/600 = e^(6k)

solve for k
k=0.80472

so
A = 600 * e^(-2*0.80472)=120

so P(0) = 120

b)
P(t) = 120 e^(0.80472*t)

c) P(3)=120 e^0.80472*3=1342

d)
dP/dt = 120*0.80472*e^(0.80472*3)=1079.6

e) 200000=120*e^(0.8047*t)

t=9.219

20)

first box 2560
second box 2560

third 2560

so k = .1*ln(3040/2560)=0.0172

so P(t) = 2560*e^(0.0172*t)

then
so 1995 was 45 years later so t=45

P(55) = 2560*e^(0.0172*45)=5551

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