the following function gives the U.S population t years after 1800. In what year will the population reach 144.7 million people? P(t)=189.4/1+34.74e^-0.031476t

Question

I am unsure on where I am suppose to pluggin the numbers.  For instance I think that I should take 2016-1801 to obtain the number of years and plug my answer to P(t).  However the 144.7 million I am unsure if I plug that number to 34.74e^-0031476t.

 

Please help

Answer 1

Hi Alina

 

Assuming the function P(t) is defined in millions, you must substitute 144.7 for P(t) to get

                               144.7=189.4/(1+34.74e^-0.031476t)  and solve for t.

First you have to multiply both sides by the denominator on the right side of the equation and divide by 144.7 to get

 

                      1+34.74e^-0.031476t = 189.4/144.7

Subtract 1 from both sides and divide by 34.74 to get

 

                         e^-0.031476t = ((189.4/144.7) - 1)/34.74 = .008892199

Take the natural log of both sides to get

                        -0.031476t - ln(.008892199)=-4.722580893

Now divide by -0.031476 on both sides to get 

 

                   t = 150.0375. Since the function is defined for the population after 1800, you have to add 150 to 1800 to get the year 1950.