The population of Austinberg on January 1, 2000 was 1.4 million
The Correct Answer and Explanation is:
The correct answer is B) p = 1.4(0.88)ᵗ.
This problem describes a situation of exponential decay, which occurs when a quantity decreases by a consistent percentage over regular time intervals. The general formula for exponential decay is p = P(1 – r)ᵗ, where ‘p’ is the final population, ‘P’ is the initial population, ‘r’ is the rate of decay expressed as a decimal, and ‘t’ is the number of time periods, in this case, years.
First, we must identify the values provided in the problem. The initial population of Austinberg, P, is 1.4 million. The problem states that the population variable ‘p’ is already in millions, so we can use the number 1.4 for P in our equation.
Next, we identify the rate of decrease, ‘r’. The population has decreased at a rate of 12% each year. To use this percentage in our formula, we must convert it into a decimal by dividing by 100. So, r = 12% = 12 / 100 = 0.12.
The core of the exponential decay formula is the decay factor, which is calculated as (1 – r). This factor represents the proportion of the population that remains from one year to the next. Since the population decreases by 12%, it means that 100% – 12% = 88% of the population remains each year. In decimal form, this is calculated as 1 – 0.12 = 0.88. This value, 0.88, is the base of the exponent in our equation.
Finally, we can assemble the complete equation by substituting the initial population and the decay factor into the formula:
p = P(1 – r)ᵗ
p = 1.4(0.88)ᵗ
This matches option B. The other options are incorrect. Option A incorrectly uses the rate of decay (0.12) as the decay factor. Options C and D model exponential growth because their bases (1.88 and 1.12) are greater than one, indicating an increase in population rather than the stated decrease
