Assume that the number of bacteria follows an exponential growth model P(t)=Pe^{kt}? The count in the bacteria culture was 500 after 15 minutes and 2000 after 40 minutes. What was the initial size of the culture?

Question

7864 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-09-30T17:48:12

The initial size of the culture #P(0)=125root(5)4~~165.#

#P(t)=Pe^(kt)............................(star)#

#"When "t=15, P(15)=500 rArr Pe^(15k)=500...........(1)#

#when t=40, P(40)=2000 rArr Pe^(40k)=2000.............(2)#

#(2)-:(1) rArr (Pe^(40k))/(Pe^(15k))=2000/500 rArr e^(25k)=4.#

#rArr (e^(25k))^(1/25)=4^(1/25) rArr e^k=4^(1/25)............(3)#

Now, rewriting #(1)" as, "P(e^k)^15=500", and now, using "(3),#

we get, #P(4^(1/25))^15=500 rArr P(4^(3/5))=500#

Hence, #P=500(4^(-4/5)).................(4)#

#(3),&, (4)", together with (star) "rArr P(t)=500(4^(-4/5))4^(t/25)#

Finally, to get the initial size of the culture, P(0) , we have to take

#t=0# in this last eqn., so that,

#P(0)=500(4^(-4/5))4^(0/25)=500(4^(-4/5))=((500)(4)(4^(-4/5)))/4#

#:. P(0)=(500/4)root(5)4=125root(5)4~~(125)(1.3195)=164.9375#

#:. P(0)~~165.#