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? () weeks

Answer 1

Answer:

The answers to the question are

• How long did it take the population to double? Answer = one week
• How long did it take the population to triple?  Answer = 1.58 weeks
• When were there 10, 000 ants on board? Answer = 5.06 weeks
• How long into the voyage were there 200 ants per anteater? Answer = 11.38 weeks

Step-by-step explanation:

To solve the question we use a similar analogy of half life calculation

Therefore we have

$N_{(t)} =N_{(0)}(2 )^{\frac{1}{t} }$  where

$N_{(t)} =$ Number of ants after time t

$N_{(0)} =$ Starting number of ants

Therefore

600 = 300 × $2^{\frac{1}{t} }$

$2^{\frac{1}{t} }$ = 2

㏑ $2^{\frac{1}{t} }$ = ㏑2

$\frac{1}{t}$㏑2 = ㏑2

$\frac{1}{t}$ = 1 and t = 1 week

It took one week for the population of the ants to double

For the population to triple, we have

3×300 = 300× $2^{\frac{x}{1} }$

900 = 300× $2^{\frac{x}{1} }$

$2^{\frac{x}{1} }$ =3

x㏑2 =㏑3 or x = $\frac{ln3}{ln2}$ = 1.58 weeks

There were 10000 ants onboard after

10000 = 300× $2^{x }$

or  $2^{x }$ = $\frac{100}{3}$ and x = (㏑$\frac{100}{3}$ )/㏑2  = 5.06 weeks

For the anteaters we have

Time for the anteaters to double = 3.2 weeks

Therefore t in the equation $N_{(t)} =N_{(0)}(2 )^{\frac{1}{t} }$ = 3.2 weeks

and the length of time it took for the population of ant eaters to grow to 200 is given by

200 = 17×$2^{\frac{x}{t} }$ =17×$2^{\frac{x}{3.2} }$

From which we have

$\frac{x}{3.2}$ ㏑2 =  ㏑(200/17)  

x = 3.2×3.56 = 11.38 weeks