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 population
r is the annual growth rate
t 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]
