A country has a food deficit of 10%. Its population grows continuously at the rate of 3% per year. Its annual food production every year is 4% more than that of the last year Assuming that the average food requirement per person remains constant, prove that the country will become self-sufficient in food after n years, where n is the smallest integer bigger than or equal to `(log_e 10-log_e 9)/((log_e 1.04)-0.03)`

Explanation

Let `P_(0)` be the initial population of country and P be the population of country in year t. Then,
`(dP)/(dt)`= rate of change of population =`3/100P=0.03P`
`therefore` Population of P at the end of n years is given by
`int_(P_(0))^(P) (dP)/(P)= int_(0)^(P)0.03dt`
or `inP-"ln"P_(0)=(0.03)n`
or `"ln"P="ln"P_(0)+(0.03)n`.............(1)
If `F_(0)` is its initial food production and F is the food production in year n, then
`F_(0) = 0.9P_(0)`
and `F=(1.04)^(n)F_(0)`
or `InF=n"ln"(1.04)+"ln"F_(0)`...............(2)
The country will be self-sufficient if `FgeP`
or `"ln"Fge"ln"P`
or `n "ln"(1.04)+"ln"F_(0)ge"ln"P_(0)+(0.03)n`
or `nge("ln"P_(0)-"ln"F_(0))/("ln"(1.04)-(0.3))=("ln"10-"ln"9)/("ln"(1.04)-0.03)`
Hence, `nge("ln"10-"ln"9)/("ln"(1.04)-0.03)`
Thus,the least integral values of the year n, when the country becomes self-suffcient, is the smallest integer greater than or equal to `("ln"10-"ln"9)/("ln"(1.04)-0.03)`