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 (in 10 - in 9) / (in (1.04)- 0.03).
Answers
Given : 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.
average food requirement per person remains constant
country will become self-sufficient in food after n years,
To Find : Show that n ≥ ( ln 10 - ln 9)/( ln(1.04) -0.03))
Solution:
country has food deficit of 10%
Let say Food requirement = F
Food available = F - (10/100)F = 0.9F
population grows continuously at a rate of 3% per year ( its not compounded annually but continuous)
=> dP/dt = 0.03P
=> dP/P = 0.03dt
=> ln P = 0.03t + C
at t = 0 P = F
=> ln F = 0 + C
=> ln P = 0.03t + ln F
=> ln(P/F) = 0.03t
=> P = Fe^(0.03t)
Food requirement after n Years = Fe^(0.03n)
production every year is 4% more than that of the last year ( this is not continuous but compounded every year )
Food available after n Years = 0.9F (1 + 4/100)ⁿ
= 0.9F(1.04)ⁿ
country will become self-sufficient
=> 0.9F (1.04)ⁿ ≥ Fe^(0.03n)
=> 0.9 (1.04)ⁿ ≥ e^(0.03n)
Taking ln (natural log ) both sides
=> ln (0.9) + n ln(1.04) ≥ 0.03n
=> n ( ln(1.04) -0.03)) ≥ - ln (0.9)
=> n ( ln(1.04) -0.03) ≥ - ln (9/10)
=> n ( ln(1.04) -0.03) ≥ - (ln 9 - ln 10)
=> n ( ln(1.04) - 0.03)) ≥ ln 10 - ln 9
=> n ≥ ( ln 10 - ln 9)/( ln(1.04) - 0.03)
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