Question

A ship embarked on a long voyage. At the start of the voyage, there were 300 ants in the cargo hold of the ship. One week into the voyage, there were 600 ants. Suppose the population of ants is an exponential function of time. (Round your answers to two decimal places.) How long did it take the population to double? weeks How long did it take the population to triple? weeks When were there 10, 000 ants on board? weeks There also was an exponentially-growing population of anteaters on board. At the start of the voyage there were 17 anteaters, and the population of anteaters doubled every 3.2 weeks. How long into the voyage were there 200 ants per anteater?

Answer 1

Answer:

It takes 1 week for the ant population to double

It takes 1.58 weeks for the ant population to triple

It takes 5.06 weeks for the ant population to reach 10000

It takes 5.09 weeks for the ant population to be 200 times the ant eater population.

Step-by-step explanation:

It takes only 1 week for the population to double from 300 to 600

We can model the population of ant (or anteater) as the following:

$p = ae^{kt}$

Where a = 300 is the initial population at t = 0

When t = 1, P = 600

$600 = 300e^{1k}$

$e^k = 2$

k = ln2 = 0.693

When the population tripled, p/a = 3

$e^{kt} = 3$

$kt = ln 3 = 1.1$

t = 1.1/k = 1.1/0.693 = 1.58 weeks.

When there are 10000 ants on board, p = 10000:

$300e^{kt} = 10000$

$e^{kt} = 10000/300 = 33.33$

$kt = ln33.33 = 3.51$

t = 3.51 / k = 3.51 / 0.693 = 5.06 weeks.

Similarly for anteater, at t = 0 there are 17 of them so A = 17. We can solve for their K parameter if the population doubled after 3.2 weeks

$e^{3.2K} = P/A = 2$

$3.2K = ln2$

$K = ln2/3.2 = 0.2166$

At the time there are 200 ants per anteater

$p = 200P$

$300e^{kt} = 200*17e^{Kt}$

$e^{kt - Kt} = 200*17/300$

$e^{0.693t - 0.2166t} = 11.33$

$e^{0.4765t} = 11.33$

$0.4765t = ln11.33$

$t = ln11.33/0.4765 = 5.09$ weeks