Question 1088830: 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 4 days? How long is it until there are 90,000 mosquitoes?
What is the size of the colony after 4 days?
Approximately __3842____ mosquitoes.
(Do not round until the final answer. Then round to the nearest whole number as needed.)
How long is it until 90,000 mosquitoes are in the colony?
________ days.
(Do not round until the final answer. Then round to the nearest tenth as needed.)
I got the first but I need help with the second answer
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! i believe the formula for uninhibited growth is:
f = p * e ^ (r * n)
f is the future value
p is the present value
r is the growth rate per time period
n is the number of time periods
you started with 1000 mosquitos and that rose to 1400 after 1 day.
f = 1400
p = 1000
r = what you want to find.
n = 1 day
formula becomes 1400 = 1000 * e ^ (r * 1)
this simplifies to 1400 = 1000 * e ^ r
divide both sides of this equation by 1000 to get:
1.4 = e ^ r
take the natural log of both sides of this equation to get:
ln(1.4) = ln(e ^ r)
this becomes ln(1.4) = r * ln(e) which becomes ln(1.4) = r
you get r = ln(1.4) = .3364722366
the continuous growth rate is .3364722366 per day.
to confirm, replace g in the original equation to get:
1000 * e^.3364722366) = 1400
after 44 days, the size of the colony would be 1000 * e^(.3364722366 * 44) which is equal to 2,689,264,815.
that'a a lot of mosquitos.
to find out how long it would take for there to be 90,000 mosquitos, use the formula as shown below:
90,000 = 1000 * e^(.3364722366 * t)
divide both sides of the equation by 1000 to get:
90,000 / 1000 = e^(.3364722366 * t).
take the natural log of both sides of this equation to get:
ln(90,000 / 1000) = ln(e^(.3364722366 * t) which becomes:
ln(90) = .3364722366 * x * ln(e) which becomes:
ln(90) = .3364722366 * x
solve for x to get x = ln(90) / .3364722366 = 13.37349469
the colony would grow to 90,000 in 13.37349469 days.
1000 * e^(.3364722366 * 13.37349469) = 90,000
using this formula, after 4 days, the number of mosquitos would be:
1000 * e^(.364722366 * 4) = 3841.6 which can be rounded to 3842.
you are correct there.
i believe you could also have used a discrete compounding formula instead of a continuous compounding formula to get the same answer.
1000 to 1400 in 1 day is a growth rate of .4 per day.
the formula for compound interest growth is f = p * (1 + r) ^ n
f is the future value
p is the present value
r is the growth rate per time period
n is the number of time periods.
the formula becomes 1400 = 1000 * (1 + r) ^ 1
solve for r to get r = 1400 / 1000 - 1 which is equal to .4
in 4 days, the population would go from 1000 to 1000 * (1.4)^4 = 3841.6 which can be rounded to 3842.
you get the same results whether you use the continuous compounding formula or the discrete compounding formula.
to solve for the number of days when the colony reaches 90,000, the discrete compound formula would become:
90,000 = 1000 * (1.4) ^ n
divide both sides of this equation by 1000 to get:
90 = (1.4) ^ n
take the log of both sides of this equation to get:
log(90) = log((1.4) ^ n)
this is equivalentto:
log(90) = n * log(1.4)
solve for n to get n = log(90) / log(1.4) = 13.37349469
this is the same answer as was derived using the continuous compounding formula.
continuous compounding formula is f = p * e ^ (r * n)
discrete compounding formula is f = p * (1 + r) ^ n
f is the future value
p is the present value
r is the growth rate per time period.
n is the number of time periods.
note that the value of r is different, but everything else will be the same.
in the continuous compounding formula, r = .3364722366 per day
in the discrete compounding formula, r = .4 per day.
when you solve for the value of the exponent, you can use the natural log function or the regular log function of your calculator.
either one will get you the correct answer.
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