Question 1123615: A biologist recorded a count of 628 bacteria present in a culture after 10 minutes and 1357 bacteria present after 20 minutes.
To the nearest whole number, what was the initial population?
Found 3 solutions by josgarithmetic, Theo, greenestamps:
Answer by josgarithmetic(39618)   (Show Source):
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! there were 628 bacteria after 10 minutes and 1357 bacteria after 20 minutes.
the formula for bacteria growth or decay is usually f = p * e^(rt)
if r is positive, you have growth.
if r is negative, you have decay.
f is the future value
p is the present value
e is the scientific constant of 2.718182828.....
t is the number of time periods.
you are given that the bacteria count is 628 after 10 minutes and 1357 after 20 minutes.
that means the bacteria grew from 628 to 1357 in 10 minutes.
that gets you f = 1357 and p = 628 and t = 10 minutes.
plug that in the formula to get 1357 = 628 * e^(10 * r)
divide both sides of that equation by 628 to get 1357 / 628 = e^(10 * r)
take the natural log of both sides of this equation to get ln(1357/628) = ln(e^(10r))
since ln(e^10r) = 10r * ln(e), and since ln(e) = 1, the equation becomes:
ln(1357/628) = 10r
solve for r to get r = ln(1357/628)/10 = .0770491493
confirm this solution is correct by replacing r in the original equation with this value to get:
1357 = 628 * e^(.0770491493 * 10)
this results in 1357 = 1357, confirming the solution is correct.
you now know the interest rate per time period using the continuous compounding formula of f = p * e^(rt)
since the bacteria count after 10 minutes is 628, this formula becomes:
628 = p * e^(.0770491493 * 10)
solve for p to get p = 628 / e^(.0770491493 * 10) = 290.6293294.
this can be rounded off to 291.
this formula can be graphed as shown below:
from the graph, you can see that, after 10 minutes it's at 628 and after 20 minutes it's at 1357 and it goes up rapidly from there.
your solution is that the initial population was 291 bacteria rounded to the nearest whole number.
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
Both of the responses you have received so far are valid; and they demonstrate general methods for solving problems like this.
But with the given information, you don't need logarithms or exponential functions.
The given information tells you the factor by which the population increased between 10 minutes and 20 minutes; the population increased by the same factor in the first ten minutes.
1357/628 = 2.160828....
628/2.160828 = 290.629...
To the nearest whole number, the initial population was 291.
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