I have no idea how to solve this

Question

In 2016 the United States had a population of about 323 million people and was growing at 0.7 %. Use an explicit exponential model to predict what year the U.S. population will reach 400 million people 

Answer 1

Use the general equation P_final = P_initial e^rt

P_final = 400 million

P_initial = 323 million

r = rate of growth = .007 (note: 0.7% in decimal form is .007 assumed to be annual growth -- per year)

t = number of years

400,000,000 = (323,000,000) e^.007t; now divide each side by 323,000,000 to isolate the e term

400,000,000/323,000,000 = e^.007t we can simplify the left side

400/323 = e^.007t now we take the natural log (ln) of both sides,

ln (400/323) = ln e^.007t using the power rule of logarithms, gives us

ln (400/323) = (.007t) ln e the natural log (ln) of e equals 1, therefore we have

ln (400/323) = .007t divide both sides by .007, gives us

t = (1/.007) ln (400/323) =~ 30.5 years

Therefore 2016 plus 30.5 years the population will reach 400 million =>>>>> sometime in the year 2046

Answer 2

Use the continuous growth formula A=Pe^rt to solve this. Some end amount A is equal to the initial amount, or principal P, times the constant e, which is a magical number that models naturally occurring growth, to the power of the rate multiplied by times compounded (r·t).

The rate of .7% converts to .007 as a decimal, so we'll plug in all the information given in the question:

400=323e^.007·t

Divide both sides by 323 to get the e^.007·t by itself:

1.23839 = e^.007·t

take the natural log of both sides to get rid of the e:

ln 1.23839 = ln e^.007·t

Using the exponent property of logarithms, we get

ln 1.23839 = .007t·ln e

And since ln is a log with a base of e, ln e = 1. So

ln 1.23839 = .007t

And t = (ln 1.23839)/7 . Plug that in your calculator, preferably using the original decimal your calculator found instead of the rounded one you wrote down, and t is about 30.5.

30.5... years after 2016 puts the population at 400 million somewhere in the year 2046.

Hope that helps! Exponential equations are cool!