Question

The count in a bacteria culture was 900 after 15 minutes and 1900 after 35 minutes. Assuming the count grows exponentially, What was the initial size of the culture? Find the doubling period. Find the population after 75 minutes. When will the population reach 15000. You may enter the exact value or round to 2 decimal places. Get help: Video

Answer 1

Answer:

• initial size: 514
• doubling time: 18.55 minutes
• 75 minute population: 8468
• time to 15000: 90.30 minutes

Step-by-step explanation:

The culture has a growth factor of 1900/900 = 19/9 in 20 minutes. Using the known value at t=15 minutes, one way to write the exponential function is ...

  p(t) = 900·(19/9)^((t -15)/20)

a) The initial size is p(0) = 900·19/9^(-15/20) ≈ 513.88 ≈ 514

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b) The doubling period is the time for the growth factor to obtain a value of 2:

  2 = (19/9)^(t/20)

  log(2) = (t/20)log(19/9)

  t = 20log(2)/log(19/9) ≈ 18.55 . . . minutes

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c) After 75 minutes, the population is ...

  p(75) = 900·(19/9)^((75 -15)/20 = 900·(19/9)^3 ≈ 8467.90 ≈ 8468

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d) The population will reach 15000 when ...

  15000 = 900·(19/9)^((t-15)/20)

  20·log(50/3)/log(19/9) = t -15

  t = 15 + 20·log(50/3)/log(19/9) ≈ 90.30397 ≈ 90.30 . . . minutes

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Comment on answer values

It's not clear what answers are supposed to be rounded to 2 decimal places. We have chosen to round times and give the nearest integer for populations. The decimal values of populations are shown, just in case.

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A graphing calculator can answer or provide a check. Here, we have found the zero of p(t)-15000 to determine the time to that milestone population.