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A country has a food deficit of `10%`. Its population grows continuously at a 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 `(ln10-ln9)/(ln(1.04)-0.03)` |
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Answer» 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)` |
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