A bacteria culture starts with 200 bacteria and grows at a rate proportional to its size. After 6 hours there will be 1200 bacteria (1) Express the population after I hours as a…

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asked Jan 28, 2020 82.8k views

1 Answer

Cyroxis · Answer 1 · 2020-02-04T08:13:31+0000

Answer:

We are given that the rate of change is proportional to its size S

So,
$(dS)/(dt) \propto S$

(dS)/(dt) = kS

(dS)/(S) = kdt

Integrating both sides

$\log(S)= kt + log c$

(S)/(S_0)=e^(kt)

S=S_0 e^(kt)

S is the population after t hours

S_0 is the initial population

Now we are given that After 6 hours there will be 1200 bacteria

1200=200 e^(6k)

6=e^(6k)

$6^{(1)/(6)=e^(k)$

So,
$S=200 * 6^{(t)/(6)$

a)Now the population after t hours as a function of t;
$S=200 * 6^{(t)/(6)$

b) What will be the population after 7 hours?

Substitute t = 7 hours

A bacteria culture starts with 200 bacteria

$S=200 * 6^{(7)/(6)}$

c) How long will it take for the population to reach 1750 ?

$1750=200 * 6^{(t)/(6)$

$(1750)/(200) =6^{(t)/(6)$

$8.75 =6^{(t)/(6)$

t=7.26

So, it will take 7.2 hours for the population to reach 1750