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.
