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 at $$t=0$$. The number of bacteria is increased by $$20$$% in $$2$$ hours. If population of bacteria is $$2000$$ after $$kloge65hours$$, then $$kloge22is$$ equal to

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 at $$t=0$$. The number of bacteria is increased by $$20$$% in $$2$$ hours. If population of bacteria is $$2000$$ after  $$kloge65hours$$, then  $$kloge22is$$ equal to

Infinity Learn · Accepted Answer

Let x be the number of bacteria present at time t, it $$givesdxdt=$$$$kx$$            Integrate both sides ⇒$$lnx=kt+c$$,            The initial condition $$is0$$,$$1000$$            $$Hencec=ln1000$$            At t $$=$$ $$2$$,$$x=1000+1000$$×$$20100=1200$$            Log $$1200=2k1+log1000$$⇒$$2k1=log12001000=log65$$⇒$$k1=12log65$$   ∴$$logx=12log65t+log1000$$                                                        Given        $$log2000=12log65$$.$$k+log1000log65$$    ⇒$$k=2log2$$                 $$kloge22=22=4$$

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 at t=0. The number of bacteria is increased by 20% in 2 hours. If population of bacteria is 2000 after kloge65hours, then kloge22is equal to

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