Question

The population of a certain country in 1997 was 288 million people. In​ addition, the population of the country was growing at a rate of 0.9​% per year. Assuming that this growth rate​ continues, the model Upper P (t )equals 288 (1.009 )Superscript t minus 1997 represents the population P​ (in millions of​ people) in year t. According to this​ model, when will the population of the country reach ​(a) 305 million​ people? ​(b) 395 million​ people?

Answer 1

Answer:

a) In the year 1998 with 3 days, 2 hours, 44 minutes and 18.1006141 seconds

b) In the year 1998 with 19 days, 18 hours, 5 minutes and 35.10875472 seconds

Step-by-step explanation:

a) To know the time when the population will be 305 million people, we need to isolate the variable t in the equation, $P(t)=288(1.009)^{t-1997}$

So we isolate t with the property of logarithms that allow us go down the exponent, applying in both sides of the equation

For this subsection the value of P is equal to 305 million people

$Ln(305)=Ln[(288)(1.009)]^{t-1997}$

$Ln(305)=(t-1997)*Ln[(288)(1.009)]$

Now, we can isolate the value of t

$\frac{Ln(305)}{Ln[(288)(1.009)]} =t-1997$

$\frac{Ln(305)}{Ln[(288)(1.009)]}-1997 =t$

$t=1998.008531776402$

So to know the exact date we multiply the number after the point, that is 0.008531776402 for the number of days that have 1 year, equal to 365 days

0.008531776402*365= 3.114098387 days

then the number after the point, that is 0.114098387 will be multiply for the number of hours that have 1 day

0.114098387*24= 2.738361282 hours

then the number after the point, that is 0.738361282 will be multiply for the number of minutes that have 1 hour

0.738361282*60= 44.3016769 minutes

Finally, the number after the point, that is 0.3016769 will be multiply for the number of seconds that have 1 minute

0.3016769*60= 18.1006141 seconds

And we obtain that the time, when the population of the country is 305 million people, is in the year 1998 with 3 days, 2 hours, 44 minutes and 25.152 seconds

b) For calculate the time when the population is 395 million people, we do the same process we did in the subsection a)

$Ln(395)=Ln[(288)(1.009)]^{t-1997}$

$Ln(395)=(t-1997)*Ln[(288)(1.009)]$

Now, we can isolate the value of t

$\frac{Ln(395)}{Ln[(288)(1.009)]} =t-1997$

$\frac{Ln(395)}{Ln[(288)(1.009)]}-1997 =t$

$t=1998.05412021527$

0.05412021527*365= 19.75387857 days

0.75387857 *24= 18.09308577 hours

0.09308577*60= 5.585145912 minutes

0.585145912*60= 35.10875472 seconds

And we obtain that the time, when the population of the country is 395 million people, is in the year 1998 with 19 days, 18 hours, 5 minutes and 35.10875472 seconds