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)`

Explanation

Let `X_(0)` be initial population of the country and `Y_(0)` be its
initial food production. Let the average consumption be
a unit. Therefore, food requird initially `aX_(0).` It is given
`Y_(p)=aX_(0)(90/100)=0.9 aX_(0) …(i)`
Let X be the population of the country in year t.
Then, `(dX)/(dt)=` Rate of change of population
`=3/100X=0.03X`
`rArr (dX)/X=0.03dt rArr int (dX)/X=int0.03dt`
`rArr log X = 0.03 t+c`
`rArr X=A cdote^(0.03 t), where A=e^(c)`
`At t=0, X=X_(0), thus X_(0)=A`
`therefore X=X_(0)e^(0.03 t)`
Let Y be the food production in year t.
Then, ` Y=Y_(0)(1+4/100)^(t)=0.9aX_(0) (1.04)^(t)`
`therefore Y_(0)=0.9 aX_(0)` [from Eq. (i)]
Food consumption in the year t is `aX_(0)e^(0.03 t).`
Again, `Y-Xge0` [given]
`rArr 0.9X_(0)a (1.04)^(t)gt a X_(0) e^(0.03 t)`
`rArr (1.04)^(t)/e^(0.03 t) gt 1/0.9 =10/9.`
Taking log on both sides, we get
`t[log (1.04)-0.03] ge log 10-log9`
`rArr t ge (log10-log9)/(log(1.04)-0.03)`
Thus, The least integral values of the year n, when the
country becomes self-sufficient is the smallest integer
greater than or equal to `(log10-log9)/(log(1.04)-0.03).`