Question 920739: Find an exponential growth model for the following scenario. A bacterial strain was measured to have a total population of 400 after 2 hours. After 5 hours the population was 900. Use this model to predict the population after 10 hours.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! i believe the exponential growth model equation is f = p * e^(rt)
f is the future value
p is the present value
e is the scientific constant of 2.71828...
r is the growth rate per time period.
t is the number of time periods.
in your problem, the bacterial grow from 400 to 900 in 3 hours.
the population is 400 after 2 hours and the population is 900 after 5 hours.
that's 3 hours in between.
your equation becomes:
900 = 400 * e^(3r)
f = 900
p = 400
t = 3 hours
r = what you want to solve for.
divide both sides of this equation by 400 and you get:
900/400 = e^(3r)
simplify to get 9/4 = e^(3r)
take the natural log of both sides of this equation to get:
ln(9/4) = ln(e^(3r))
since ln(e^(3r)) equals 3r * ln(e) and ln(e) = 1, you get:
ln(9/4) = 3r
divide both sides of the equation by 3 to get:
ln(9/4) / 3 = r
solve for r to get r = ln(9/4) / 3 = .2703100721
confirm by replacing r with .2703100721 in the original equation to get:
900 = 400 * e^(.2703100721*3) which becomes:
900 = 900.
this confirms the solution is good.
you are asked to find how many bacteria after 10 hours.
the formula for that is the same general formula of f = p * e^(rt)
you can choose p to be 400 or 900
if you choose 400, then t = 10-2 = 8 hours
if you choose 900, then t = 10 - 5 = 5 hours
you should get the same answer.
i'll do both to show you.
when p = 400
t = 10-2 = 8
r = .2703100721
f = what you want to find.
you get:
f = 400 * e^(.2703100721 * 8) = 3477.069512
when p = 900
t = 10-5 = 5
r = .2703100721
f = what you want to find.
you get:
f = 900 * e^(.2703100721 * 5 = 3477.069512
you're good either way.
|
|
|
