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Calculus IChapter 1: ReviewSection 1-8

Logarithm Functions — Question 9

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Question 9

A bacteria culture grows according to the function N(t)=N0ektN(t) = N_0 e^{kt} where N0N_0 is the initial population, kk is a constant, and tt is the time in hours.

Suppose a culture starts with 500 bacteria and grows to 2000 bacteria in 6 hours.

  • (a) Find the growth rate kk.

  • (b) Use your result to estimate the population after 10 hours.

  • (c) After how many hours will the population reach 10,000?

Original worksheet page 1: question and worked solution for 1-8-009
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Question 9 - Solution

Growth rate. 2000=500e6k2000=500e^{6k} gives

k=ln⁡46≈0.231049hr−1.\boxed{k=\frac{\ln4}{6}\approx 0.231049\ \mathrm{hr}^{-1}}.

Population after ten hours.

N(10)=500⋅410/6≈5039.68,N(10)=500\cdot4^{10/6}\approx 5039.68,

so the estimated whole-number population is 5040\boxed{5040}.

Time to reach 1000010000.

500ekt=10000⇒t=6ln⁡20ln⁡4≈12.97hours.500e^{kt}=10000\quad\Longrightarrow\quad \boxed{t=\frac{6\ln20}{\ln4}\approx 12.97\ \text{hours}}.

Substitution into the exact model verifies both target populations.

Original worksheet page 2: question and worked solution for 1-8-009

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