Suppose that the Russian population is declining by 0.48% each year, and continues to decline at this rate every year.
a.) what is the overall 'growth' factor (really a decline factor) for 40 years of population decline? As a percentage of the current population, what will the Russian population be 40 years from now?
b.) by what overall percentage will the population decline over the next 40 years?
c.) look again at part b of this question. Suppose that someone makes the argument: "if the Russian population declines by 0.48% each year, then after 210 years there won't be any population left, since (0.48%)(210) is greater than 100%" What is wrong with this argument?
well, clearly the population, starting from an initial value of P, is
p(t) = P(0.9952^t)
after t years. So,
p(40)/P = 0.9952^40 = 0.8249
That makes a decrease of 17.51%
I think you can now refute the argument about zero population.
To calculate the answers step-by-step:
a.) To find the overall decline factor, we can convert the annual decline rate of 0.48% to a decimal by dividing it by 100: 0.48/100 = 0.0048.
1. Subtract the decline rate from 1: 1 - 0.0048 = 0.9952.
2. Raise the result to the power of the number of years (40): 0.9952^40 = 0.819.
So, the overall decline factor for 40 years of population decline is 0.819.
To calculate the population 40 years from now, we multiply the current population by the decline factor:
Population 40 years from now = Current population * Overall decline factor
= Current population * 0.819.
b.) To calculate the overall percentage decline over the next 40 years, we subtract the overall decline factor from 1 and multiply by 100:
Overall percentage decline = (1 - Overall decline factor) * 100.
c.) The argument that after 210 years there won't be any population left due to the decline rate of 0.48% each year is incorrect.
0.48% per year means a decline factor of 0.9952.
If we raise this decline factor to the power of 210 (the number of years), we get 0.9952^210 = 0.0956.
This indicates that after 210 years, the population would be reduced to 9.56% of the initial population, not 0%.
a.) To find the overall decline factor for 40 years of population decline, we need to multiply the decline rate (-0.48%) by the number of years (40).
Decline factor = (1 - decline rate)^number of years
Decline factor = (1 - 0.0048)^40
Using a calculator, we can find the decline factor to be approximately 0.812035.
To calculate the projected population 40 years from now, we multiply the current population by the decline factor:
Projected population = Current population * Decline factor
b.) To find the overall percentage decline over the next 40 years, we need to subtract the decline factor from 1 (since the decline factor represents the proportion of the population remaining) and then express it as a percentage.
Overall percentage decline = (1 - Decline factor) * 100%
Using the value of the decline factor obtained in part a, we can calculate the overall percentage decline over the next 40 years.
Overall percentage decline = (1 - 0.812035) * 100%
c.) The argument presented in this part is incorrect. The mistake lies in assuming that if a decline rate of 0.48% per year continues indefinitely, after 210 years, the population will reach zero.
However, the decline rate is a percentage of the remaining population, not the original population. Therefore, it cannot exceed 100% of the remaining population. The decline rate of 0.48% per year does not accumulate to the point where the population reaches zero after 210 years.