Population increase question

Question

According to the U.S. Census Bureau, the population of the United States in 2008 was 304 million people. In addition, the population of the United States was growing at a rate of 1.1% per year. Assuming this growth rate continues, the modelrepresents the population  (in millions of people) in year .

According to the model, when will the population be 460 million people? Be sure to round your answer to the nearest whole year.

Answer 1

460 = 304e^rt where r = 0.011 and t = years after 2008

460/304 = e(.011)t

ln(460/304) = 0.011t

t = ln(460/304)/.011 = 37.65

t = 37.65 years to reach 460 million

2008 + 38

= 2046 AD or CE = the year when population reaches 406 million

but your problem uses a slightly different model as if population doesn't grow continuously but only annually, which is "close" but clearly not how population grows, realistically.

so

460 = 304(1+ .011)^t

460/304 = 1.011^t

solve for t

1.5132 = (1.011)^t

t = the power, the exponent that makes 1.011 = 1.5132

t > 32 years using a simple calculator, but probably not much more

t = a little less than 37 years

year about 2008+37 = 2045 AD

with just annual compounding, it grows slightly less slowly than with continuous compounding