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Question

The rate of growth of bacteria in a culture is proportional to the number of bacteria present and the bacteria count is 1000 at initial time t=0. The number of bacteria is increased by 20 % in 2 hours. If the population of bacteria is 2000 after kloge(65) hours, then (kloge2)2 is equal to :

A
4
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B
2
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C
16
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D
8
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Solution

The correct option is A 4
Let x be the number of bacteria at time t.
dxdtx
dxdt=λx
x1000dxx=t0λ dt
lnxln1000=λt
ln(x1000)=λt
Put t=2, x=1200
ln(1210)=2λλ=12ln(65)

Now, ln(x1000)=t2ln(65)
x=1000 et2ln(65)
Given, x=2000 at t=kloge(65)
2000=1000 ek2ln(6/5)×ln(6/5)
2=ek/2
ln2=k2
kln2=2
(kln2)2=4

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