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Question 1140192: Question Help
The population of a colony of mosquitoes obeys the law of uninhibited growth. If there are 1000 mosquitoes initially and there are 1400 after 1 day, what is the size of the colony after
3 days? How long is it until there are 70,000 mosquitoes?
Found 2 solutions by Theo, greenestamps:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! population growth or decay uses the continuous compounding formula.
that formula is f = p * e ^ (r * n)
f is the future value
p is the present value
e is the scientific constant of 2.718281828.....
n is the number of time periods
these are names i assigned to the variables.
other people might assign different names, but the formula is the same.
you are given that there are 1000 mosquitoes initially and there are 1400 after 1 day.
the formula becomes 1400 = 1000 * e ^ (r * 1)
divide both sides of the formula by 1000 to get:
1400 / 1000 = e ^ (r * 1)
take the natural log of both sides of the formula to get:
ln(1400 / 1000) = ln(e ^ (r * 1))
since ln(e ^ (r * 1)) is equal to r+ * 1 * ln(e) and since ln(e) is equal to 1, then the formula becomes:
ln(1400 / 1000) = r.
solve for r to get r = .3364722366
replace r in the original equation to confirm the solution is good.
1400 = 1000 * e ^ (r * 1) becomes 1400 = 1000 * e ^ (.3364722366 * 1) which becomes 1400 = 1400, confirming the solution is correct.
after 3 days, the size of the colony will be based on f = 1000 * e ^ (.3364722366 * 3) which becomes f = 2744
to find out how long until there are 70,000 mosquitoes, the formula becomes 70,000 = 1000 * e ^ (.3364722366 * n)
divide both sides of the equation by 1000 to get:
70,000 / 1000 = e ^ (.3364722366 * n)
take the natural log of both sides of the equation to get:
ln(70,000 / 1000) = ln(e ^ (.3364722366 * n))
since ln(e ^ (.3364722366 * n)) = .3364722366 * n * ln(e) and since ln(e) = 1, the equation becomes:
ln(70,000 / 1000) = .3364722366 * n
divide both sides of this equation by .3364722366 to get:
ln(70,000 / 1000) / .3364722366 = n
solve for n to get n = 12.62658484 days.
replace n in the original equation to confirm this is true.
70,000 = 1000 * e ^ (.3364722366 * n) becomes 70,000 = 1000 * e ^ (.3364722366 * 12.62658484) which becomes 70,000 = 70,000, confirming the solution is correct.
Answer by greenestamps(13198)  (Show Source):
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