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1 Pure Mathematics

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Question 34

A population of bacteria in a laboratory culture is being monitored. The number of bacteria, BBB, in the culture, t t\,t hours after the initial observation, is modelled by the equation

B=400ekt7+ekt B = \frac{400e^{kt}}{7 + e^{kt}} B=7+ekt400ekt​

where k k\,k is a constant.

a.

Find the number of bacteria in the culture at the start of the study.

[2]
b.

Given that there are 160 bacteria in the culture after 5 hours,

show that k=15ln⁡(143)\displaystyle k = \frac{1}{5}\ln\left(\frac{14}{3}\right)k=51​ln(314​).

[3]
c.

Given also that, when t=Tt = Tt=T, the number of bacteria is increasing at a rate of 25 per hour,

find the possible values of TTT, giving your answers to one decimal place.

[6]

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