Posted by Vanessa on .
A population of 500 E. coli bacteria doubles every 15 minutes. Use this information to find an expression for this population growth. Using this expression, find what the population would be in 87 minutes. Use an exponential model.
so we're supposed to use P(t)= (Po)(e)^kt
so P(o) would be 500,
and then would t be 15 or 0.25 and wowhat would k be? 2?

Calculus 
Reiny,
in P(t) = 500 e^kt
it depends whether you want to to be in minutes or hours.
let's do it for minutes, then
1000 = 500 e^15k
2 = e^15k
15k = ln2
k = (ln2)/15
if you want t to be hours then use .25 for t in
1000 = 500 e^.25K
.
.
K = ln2/.25 or K = 4ln2 
Calculus 
Vanessa,
okay, so then in 87 minutes what would it be?

Calculus 
Reiny,
let's use our first one, the one in minutes
P(87) = 500 e^(87*ln2/15)
= 500 e^4.020254
= 27857.6
is our answer reasonable??
remember it doubles every 15 min
so after 15 min > 1000
after 30 min > 2000
after 45 min > 4000
after 60 min > 8000
after 75 min > 16000
after 90 min > 32000
we could check to see if we get 32000 for 90 minutes
P(90) = 500 e^(90*ln2/15)
= 500(64) = 32000 YES!!!! 
Calculus 
Vanessa,
whao! that makes so much sense.
just a quick question, how do u know what e is?
thanks so much
<3333333333 
Calculus 
Reiny,
e is one of these very strange transcendental numbers like pi
one defn of e is
1 + 1/1! + 1/2! + 1/3! + ... to infinitity
where something like 5! = 1*2*3*4*5
or
e = Limit (1 + 1/n)^n where n approaches infinitity
The Swiss/German mathematician Euler spent a lot of time with the concept of that number and e is often called Euler's number 
Calculus 
Anonymous,
e is approximately 2.71828