Question

The population of a small town is 10,100. Its population falls by 6% per year. How many years later will the population be 5,050? (Assume that the growth rate is annualized and use the formula for compound growth: A = P(1 + r/n)ⁿᵗ. If the compounding frequency is not given, use n = 1.

a) 10 years
b) 12 years
c) 15 years
d) 20 years

Answer 1

The population of the small town decreases by 6% per year. Using the compound growth formula, it will take approximately 12 years for the population to be 5,050.

Step-by-step explanation:

To find out how many years it will take for the population of the town to reach 5,050, we can use the compound growth formula: A = P(1 + r/n)ⁿᵗ. In this case, the initial population (P) is 10,100 and the population falls by 6% per year, which means the growth rate (r) is -0.06. We want to find the number of years (t) when the population (A) is 5,050.

Plugging in the values into the formula and solving for t:

5,050 = 10,100(1 - 0.06/1)ᵗ

Divide both sides by 10,100:

0.5 = (1 - 0.06)ᵗ

Take the natural logarithm of both sides:

ln(0.5) = t * ln(0.94)

Divide ln(0.5) by ln(0.94) to solve for t:

t ≈ 11.4

Rounding up to the nearest whole number, it will take approximately 12 years for the population to be 5,050.