# Calc

The number of bacteria in a culture is increasing according to the law of exponential growth. There are 125 bacteria in the culture after 2 hours and 350 bacteria after 4 hours.

a) Find the initial population.

b) Write an exponential growth model for the bacteria population. Let t represent time in hours.

c) Use the model to determine the number of bacteria after 8 hours.

d) After how many hours will the bacteria count be 25,000?

Thank you!!

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1. let the number be n

n = a e^(kt), where a is the initial number and t is the time in hours, k is a constant

case 1, when t=2, n = 125
125 = a e^2k
case 2 , when t = 4, n= 350
350 = a e^4k

divide the two equations
350/125 = a e^4k / (a e^(2k))
2.8 = e^2k
2k = ln 2.8
k = .5148

in first equation,
125 = a e^(2(.5148))
a = 44.64

a) so initially there were 46 bacteria.

b) n = 44.64 e^(.5148t)

c) when t = 8 , n = 44.64 e^(8(.5148)) = appr. 2744

d) 25000 = 44.64 e^(.5148 t)
560.0358 = e^ .5148t
.5148t = ln 560.0358
t = 12.29 hours

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