Home
Class 12
MATHS
A country has a food deficit of 10%. Its...

A country has a food deficit of 10%. Its population ear. 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)`

Text Solution

Verified by Experts

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)`
Promotional Banner

Similar Questions

Explore conceptually related problems

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)

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)

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)

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)

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)

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)

If is known that, if the interest is compounded continuously, the principal changes at the rate equal to the product of the rate of interest per annum, and the principal. If the interest is compounded continuously at 5% per annum, in how many years will Rs. 100 double itself? At what interest rate will Rs. 100 double itself in 10 years ((log)_e2=0. 6931) How much will Rs. 1000 be worth at 5% interest after 10 years? (e^(0. 5)=1. 648)dot