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:a) In the year 1998 with 3 days, 2 hours, 44 minutes and 18.1006141 secondsb) In the year 1998 with 19 days, 18 hours, 5 minutes and 35.10875472 secondsStep-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, [tex]P(t)=288(1.009)^{t-1997}[/tex]So we isolate t with the property of logarithms that allow us go down the exponent, applying in both sides of the equationFor this subsection the value of P is equal to 305 million people[tex]Ln(305)=Ln[(288)(1.009)]^{t-1997}[/tex][tex]Ln(305)=(t-1997)*Ln[(288)(1.009)][/tex]Now, we can isolate the value of t[tex]\frac{Ln(305)}{Ln[(288)(1.009)]} =t-1997[/tex][tex]\frac{Ln(305)}{Ln[(288)(1.009)]}-1997 =t[/tex][tex]t=1998.008531776402[/tex]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 days0.008531776402*365= 3.114098387 daysthen the number after the point, that is 0.114098387 will be multiply for the number of hours that have 1 day0.114098387*24= 2.738361282 hoursthen the number after the point, that is 0.738361282 will be multiply for the number of minutes that have 1 hour0.738361282*60= 44.3016769 minutesFinally, the number after the point, that is 0.3016769 will be multiply for the number of seconds that have 1 minute0.3016769*60= 18.1006141 secondsAnd 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 secondsb) For calculate the time when the population is 395 million people, we do the same process we did in the subsection a)[tex]Ln(395)=Ln[(288)(1.009)]^{t-1997}[/tex][tex]Ln(395)=(t-1997)*Ln[(288)(1.009)][/tex]Now, we can isolate the value of t[tex]\frac{Ln(395)}{Ln[(288)(1.009)]} =t-1997[/tex][tex]\frac{Ln(395)}{Ln[(288)(1.009)]}-1997 =t[/tex][tex]t=1998.05412021527[/tex]0.05412021527*365= 19.75387857 days0.75387857 *24= 18.09308577 hours0.09308577*60= 5.585145912 minutes0.585145912*60= 35.10875472 secondsAnd 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