The population of a certain country since January 1, 1910 can be approximated by the model below, where tt is the number of years since January 1, 1910.

Question

The population of a certain country since January 1, 1910 can be approximated by the model below, where tt is the number of years since January 1, 1910.

P(t)=56.58 / 1+7.2e^−0.02706t

where P is the population of this country (in millions) tt years after January 1, 1910.

(a) What was the population of this country on January 1, 1910?

__________ million

(b) Use the function to approximate the population of this country on January 1, 1927. Round your answer to the nearest whole number.

__________ million

(c) In what year will the population reach 21 million? Round your answer to the nearest year.

__________ years

Answer 1

So the original equation is P(t)=56.58 / 1+7.2e^-0.02706t

a) The population of this country on January 1, 1910 is:

=> P(t=0)=56.58 / 1+7.2e^-0.02706(0)

=> 6.9 million.

b) the population of this country on January 1, 1927 :

t=1927-1910=17 years

=> P(t=17)=56.58 / 1+7.2e^-0.02706(17)

=> P(t=17)= 10.2

Therefore the population on January 1, 1927 will be: 10.2million.

c)the year in which the population will reach 21 million is:

P(t)=56.58 / 1+7.2e^-0.02706t

P(t)= 21

21= 56.58 / 1+7.2e^-0.02706t

Then you solve for t using any method (logarithm being the best one):

t= 53.46 years => 53 years

Therefore it will take 53 years for the population to reach 21 million

Therefore the population will reach 21 million in the year: 1963 .

Hope this helped you dear student!!

Answer 2

a) No time passes between the start date, January 1, 1910, and the date in part a. So t, representing time in years, would be zero. If we plug that into the equation, we get:

P(0) = 56.58/1 +7.2e^-0.02706(0)

P(0) = 56.58/1 +7.2(1)

P(0) = 56.58/8.2

P(0) = 6.9

The answer would be 6.9 million

b) January 1, 1927, is 17 years away from January 1, 1910, so t = 17 (years). We will once again plug this value of t into the equation.

P(17) = 56.58/1 + 7.2e^-0.02706(17)

P(17) = 56.58/1+ (4.5452)

P(17) = 56.58/5.5452...

P(17) = 10.2

The answer would be 10.2 million

c) Here, we need to plug 21 million in for P(t) since it represents population. Doing this will allow us to solve for t, or the number of years needed to get the population to 21 million people:

21 = 56.58/1 + 7.2e^-0.02607t

21(1 + 7.2e^-0.02607t) = 56.58

1 + 7.2e^-0.02607t = 56.58/21

1 + 7.2e^-0.02607t = 2.6493...

7.2e^-0.02607t = 1.6493...

e^-0.02607t = 0.2353...

-0.02607t = ln(0.2353...)

t = ln(0.2353...)/-0.02607

t = 55.49 ≈ 56

The answer would be 56 years