College Algebra

Question

The population of a colony of mosquitoes obeys the law of uninhibited growth. If there are 1000 mosquitoes initially and there are 1300 after 1​ day, what is the size of the colony after 33 ​days? How long is it until there are

50,000 mosquitoes?

Answer 1

Hello, thank you for taking the time to post your question!

To model this situation you want to use the exponential function

P(t) = P0 * e^(kt)

Plugging in the values given to us in the question that becomes

1300 = 1000 * e^(k(1)) , meaning that

k = ln(1.3) = 0.2624

so the general form of the equation that we’re using to model this situation and solve the other parts is P(t) = 1000 * e^(0.2624t)

the first part it’s after then is the value at t = 33, which will be

P(33) = 1000 * e^(0.2624 * 33) = 5,763,500, meaning that the size of the mosquito colony after 33 days will be approximately 5,763,500 mosquitos

Then to find the time to reach 50,000 mosquitos you want to plug in P(t) = 50000 and then solve for the value of t

50000 = 1000 * e^(0.2624t)

ln(50) = 0.2624t

t = 3.912/0.2624 = 14.908, meaning that it will take 14.91 days until there are 50,000 mosquitos in the population

I hope that helps you get moving in a better direction on this type of question! Feel free to reach out if you have any additional questions beyond that :)

Answer 2

Not sure "uninhibited" growth means exponential, but suppose it does.

1000a^t = y

Using the situation that in t = 1, y= 1300

we get. 1000a = 1300. so a = 1.3

Now 1000(1.3^t) = y

then if t = 33 ----> 1000(1.3³³) = 5756130.4. mosquitoes

if 1000(1.3^t) = 50,000

then 1.3^t = 50

then log 1.3^t = log 50

t log 1.3 = log 50

t =. log 50 / log 1.3

t = 14.9 days