Country A has a growth rate of 4.6% per year. The population is currently 5 comma 333,000, and the land area of Country A is 27,000,000,000 square yards. Assuming this growth rate continues and is exponential, after how long will there be one person for every square yard of land? This will happen in nothing years.
Accepted Solution
A:
Answer:After [tex]t = 189.66\ years[/tex]Step-by-step explanation:There will be one person per square yard when the number of people is equal to the number of square yards of the country.In other words, we need to know when the population of the country will be 27,000,000,000.We know that the population grows exponentially, so we use the exponential growth formula:[tex]P = p_0(1+r)^t[/tex]Where:P is the population as a function of time[tex]p_0[/tex] is the initial populationr is the annual growth ratet is time in year.The information we have allows us to conclude that:[tex]r= 4.6\%= 0.046[/tex][tex]p_0 = 5,333,000[/tex]So the exponential growth equation is:[tex]P = 5,333,000(1+0.046)^t[/tex]We want to know when P = 27, 000,000,000.So: [tex]27,000,000,000= 5,333,000(1+0.046)^t[/tex]Now we solve for t.[tex]\frac{27,000,000,000}{5,333,000} = (1+0.046)^t[/tex][tex]log(\frac{27,000,000,000}{5,333,000}) = tlog(1+0.046)[/tex][tex]t= \frac{log(\frac{27,000,000,000}{5,333,000})}{log(1+0.046)}\\\\\\t = 189.66\ years[/tex]