Algebra word problem. HELP!

Question

country A has a growth rate of 3.1 per year. The population is currently 4,539,000 and the land area of country A is 31,000,000,000 sq yards. Assuming the growth rate continues and is exponential, how long will there be one person for every sq yard of land?

Answer 1

When the population will reach 31,000,000,000 then it would be one person per sq yard

 

31,000,000,000=4,539,000(1+0.031)^t  this is the formula for exponential growth

solving this equation for t will be the answer

 

log(31,000,000,000)=log(4,539,000)+t log(1.031)

 

 10.49136=6.65696+t(0.0132587)

 

t=(10.49136-6.65696)/0.0132587=289.1988

 

It will take a little more then 289 years.

Answer 2

I assume the 3.1 should be 3.1%.  Let P(t) = the population at time t (years):

 

P(t) = (4,539,000)*(1.031)^t

 

There will be one person per square yard when the population reaches 31,000,000,000:

 

31,000,000,000 = (4,539,000)*(1.031)t

(31,000,000,000)/(4,539,000) = (1.031)t

6829.7 = (1.031)t

ln(6829.7) = t*ln(1.031)

ln(6829.7)/ln(1.031) = t

 

289.2 years = t