FORMULA BREAK DOWN

Question

suppose a certain bacteria that has been treated with an antibiotic begins to decay continuously at a rate of 2.8% per hour. How many bacteria will remain in the original sample of 400000 after 24 hours?

 

using formula : A(24)=A(0)E^RT

Answer 1

After 1 hour, you are down to (1 - 0.028)  = 0.972 times what it was at the start of the hour.

 

0.972 = (1)e^-k(1)

 

ln(0.972) = -k

 

k = -ln(0.972) ≅ 0.0283995

 

A(t) = A(0)e^-0.0283995t 

 

t = hours

A(0) = initial population = 400,000

 

Plug in t = 24 hours and get A(24).

Answer 2

The simplest way to set this up is to note that  1 - .028 = .972   So

 

A(t) =  A(0) .972^t  Where t is number of hours. 

 

Since the formula listed is in e format this equation can be rewritten as

 

A(t) =  A(0) exp [ ln( .972 t) ] =  A(0) exp[ t ln(.972 ] = A(0) exp(- .0284 t )

   This works because  ln(.972) = -.0284

 

A(24) = 400000 exp(- .0284  24) =  202325

 

Of course the formula of the first line gives the same answer.