Differential Equations 3 Question 12

12. A country has 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 ln10ln9ln(1.04)(0.03).

(2000,10M)

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Answer:

Correct Answer: 12. (x2+y2=2x) 14. log3412

Solution:

  1. Let X0 be initial population of the country and Y0 be its initial food production. Let the average consumption be a unit. Therefore, food required initially aX0. It is given

Yp=aX090100=0.9aX0

Let X be the population of the country in year t.

Then, dXdt= Rate of change of population

=3100X=0.03XdXX=0.03dtdXX=0.03dtlogX=0.03t+cX=Ae0.03t, where A=ec At t=0,X=X0, thus X0=AX=X0e0.03t

Let Y be the food production in year t.

Then, Y=Y01+4100t=0.9aX0(1.04)t

Y0=0.9aX0

Food consumption in the year t is aX0e0.03t.

Again, YX0

0.9X0a(1.04)t>aX0e0.03t(1.04)te0.03t>10.9=109.

[given]

Taking log on both sides, we get

t[log(1.04)0.03]log10log9

tlog10log9log(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 log10log9log(1.04)0.03.



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