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

Answer 2

Using exponential functions, it is found that, for the population of ants:

• It took 1 week for the population to double.
• It took 1.58 weeks for the population to triple.
• There were 10,000 ants on board after 5.06 weeks.

For the population of anteaters, 11.38 weeks into the voyage there were 200 ants per anteater.

An increasing exponential function is modeled by:

$A(t) = A(0)(1 + r)^t$

In which:

• A(0) is the initial amount.
• r is the growth rate, as a decimal.

For the population of ants, we have that:

• The initial value is 300, hence $A(0) = 300$.
• Doubled after 1 week, hence $A(1) = 600$, and this is used to find r.

$A(t) = A(0)(1 + r)^t$

$A(t) = 300(1 + r)^t$

$300(1 + r) = 600$

$1 + r = 2$

$r = 1$

Hence:

$A(t) = 300(2)^t$

It took 1 week for the population to double.

For the time it took to triple, we find t for which $A(t) = 3A(0) = 900$, hence:

$A(t) = 300(2)^t$

$900 = 300(2)^t$

$2^t = 3$

$\log{2^t} = \log{3}$

$t\log{2} = \log{3}$

$t = \frac{\log{3}}{\log{2}}$

$t = 1.58$

It took 1.58 weeks for the population to triple.

The time it took for there to be 10000 ants on board is t for which $A(t) = 10000$, hence:

$A(t) = 300(2)^t$

$10000 = 300(2)^t$

$2^t = \frac{10000}{300}$

$\log{2^t} = \log{\frac{10000}{300}}$

$t\log{2} = \log{\frac{10000}{300}}$

$t = \frac{\log{\frac{10000}{300}}}{\log{2}}$

$t = 5.06$

There were 10,000 ants on board after 5.06 weeks.

For the population of anteaters, we have that:

• Initially, there were 17, hence $A(0) = 17$.
• It doubles every 3.2 weeks, hence $A(3.2) = 2A(0)$, and this is used to find r.

$A(t) = A(0)(1 + r)^t$

$2A(0) = A(0)(1 + r)^{3.2}$

$(1 + r)^{3.2} = 2$

$\sqrt[3.2]{(1 + r)^{3.2}} = \sqrt[3.2]{2}$

$1 + r = 2^{\frac{1}{3.2}}$

$1 + r = 1.24185781207$

Hence:

$A(t) = A(0)(1 + r)^t$

$A(t) = 17(1.24185781207)^t$

The amount of time it takes for there to be 200 ants is t for which $A(t) = 200$, hence:

$A(t) = 17(1.24185781207)^t$

$200 = 17(1.24185781207)^t$

$(1.24185781207)^t = \frac{200}{17}$

$\log{(1.24185781207)^t} = \log{\frac{200}{17}}$

$t\log{1.24185781207} = \log{\frac{200}{17}}$

$t = \frac{\log{\frac{200}{17}}}{\log(1.24185781207)}$

$t = 11.38$

11.38 weeks into the voyage there were 200 ants per anteater.

A similar problem is given at https://brainly.com/question/14773454