Question 968065: i need help with this question:
The number of bacteria doubles after 3 hours. If there are N bacteria to start with, find the number of bacteria in 24 hours?
Found 2 solutions by josgarithmetic, Theo:
Answer by josgarithmetic(39620)   (Show Source):
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! it doubles every 3 hours.
this can be solved using either a continuous compounding function of f = p * e^(rn), or it can be solved using a discrete compounding formula of f = p * (1+r)^n.
r is the interest rate per time period.
n is the number of time periods.
f is the future value
p is the present value
e is the scientific constant that is equal to 2.718281828.
it's an irrational number that has an endless number of decimal digits.
most often it is shown as such because the display number of digits on the calculator has a limitation of around 9 decimal digits.
the continuous compounding function is the one most often used, but i'll show you how to do it with both.
using the continuous compounding formula, you would do the following.
the equation is f = p * e^(rn).
the number of bacterial doubles in 3 hours, so you would set:
p = 1
f = 2
n = 3
the formula of f = p * e^(rn) would become:
2 = 1 * e^(rn)
you would now need to solve for r.
you would normally divide both sides of the equation by p to get:
f/p = e^(rn) would become:
2 = e^(3r).
you would then take the natural log of both sides of the equation to get:
ln(2) = ln(e^(3r)).
since ln(e^(3r)) is equivalent to 3*ln(e), and since ln(e) is equal to 1, the equation would then become:
ln(2) = 3r.
you would then divide by 3 to get:
ln(2)/3 = r
you would then solve for r to get:
r = ln(2)/3 = .2310490602.
you can confirm by replacing r in the original equation with .2310490602.
f = p * e^(rn) would become 2 = 1 * e^(.2310490602 * 3) which would then become 2 = 2.
this would confirm the solution for r is correct.
now that you found r, you would use that to solve the problem.
go back to the same equation and do the following:
f = p * e^(rn)
set n = 24
set p = 1
set r = .2310490602
equation becomes:
f = 1 * e^(.2310490602 * 24)
solve for f to get:
f = 256
the bacteria will be 256 times the original amount in 24 hours.
the continuous compounding formula will be the one most often used, but you can solve this problem using the discrete compounding formula as well.
here's how you would solve it using the discrete compounding formula.
that formula is:
f = p * (1+r)^n
f is the future value
p is the present value
r is the interest rate per time period.
n is the number of time periods.
you would do the following:
f = 2
p = 1
n = 3
the formula of f = p * (1+r)^n would become 2 = 1 * (1+r)^3
you would divide both sides of the equation by p to get:
2/1 = (1+r)^3
note that, since p = 1, this step is not necessary, but p is not always equal to 1, so i'm showing you that step anyway.
2/1 = (1+r)^3 would become:
2 = (1+r)^3
you would raise both sides of tghe equation to the 1/3 power to get:
2^(1/3) = ((1+r)^3)^(1/3)
since ((1+r)^3)^(1/3) is equal to (1+r)^(3*1/3) which is e3qual to (1+r)^1 which is equal to 1+r, your equation would become:
2^(1/3) = 1 + r
solve for 1+r to get:
1+r = 2^(1/3) = 1.25992105
subtract 1 from that to get:
r = .25992105.
you can confirm by replacing r in th eoriginal equation to get:
2 = 1 * (1.25992105)^3 becomes 2 = 2.
this confirms the value of r is good.
now that you found r, you can use it to solve the problem.
set p = 1 and set n = 24 and set r = .25992105.
formula of f = 1 * (1+r)^n becomes:
f = 1 * (1.25992105)^24
solve for f to get:
f = 256.
you got the same answer, as you should if you did the process correctly.
the continuous compounding formula is, once again.
f = p * e^(rn)
f is the future value
p is the present value
e is the scientific constant of 2.718281828
r is the interest rate per time period.
n is the number of time periods.
the discrete compounding formuls is, once agakin.
f = p * (1+r)^n
f is the future value
p is the present value
r is the interest rate per time period.
n is the number of time periods.
in your problem, the time periods were expressed in hours.
n was therefore the number of hours.
r was therefore the interest rate per hour.
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