A culture starts with 8600 bacteria After one hour the count

A culture starts with 8600 bacteria. After one hour the count is 10,000.

(a) Find a function that models the number of bacteria n(t) after t hours. (Round your r value to three decimal places.)

(b) Find the number of bacteria after 2 hours. (Round your answer to the nearest hundred.)

(c) After how many hours will the number of bacteria double? (Round your answer to one decimal place.)

Solution

general exponential growth equation is n(t)=(n(0))e^rt

culture starts with 8600 bacteria, so n(0)=8600

=>n(t)=8600e^rt

(a)

given after one hour the count is 10000 =>n(1)=10000

=>10000=8600e^r*1

=>e^r=(10000_/8600)

=>r=ln(10000_/8600)

=>r=ln(50_/43)

=>r=0.151

(b)

n(t)=8600e^{(ln(50_/43))t}

after 2 hours

number of bacteria =n(2)

number of bacteria =8600e^{(ln(50_/43))*2}

number of bacteria =11627.9

number of bacteria =11600

(c)

number of bacteria double. =>n(t)=8600*2

=>8600e^{(ln(50_/43))t}=8600*2

=>e^{(ln(50_/43))t}=2

=>(ln(50_/43))t=ln2

=>t=(ln2)_/(ln(50/43))

=>t=4.6 hours

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