Answer:
To find the number of years it will take for the population of the colony to grow from 80,000 to 92,610 with a growth rate of 5% per annum, we can use the formula for compound interest:
Final Population = Initial Population * (1 + Growth Rate) ^ Number of Years
92,610 = 80,000 * (1 + 0.05) ^ t
Now, we can solve for t:
(92,610 / 80,000) = (1.05) ^ t
1.157625 = (1.05) ^ t
To find t, we can use logarithms:
t = log(1.157625) / log(1.05)
t ≈ 2.967
So, it will take approximately 2.967 years for the population to grow from 80,000 to 92,610 at a 5% growth rate.
Now, let's consider a 2% lower growth rate (5% - 2% = 3%). We can use the same formula to find the final population after the same time (2.967 years):
Final Population = Initial Population * (1 + Growth Rate) ^ Number of Years
Final Population = 80,000 * (1 + 0.03) ^ 2.967
Final Population ≈ 80,000 * 1.09364
Final Population ≈ 87,491.2
To find the difference in population for the same time, we can subtract the population with the lower growth rate from the population with the higher growth rate:
Difference in population = 92,610 - 87,491.2
Difference in population ≈ 5,118.8
So, the difference in population for the same time with a 2% lower growth rate would be approximately 5,118.8.