The Correct Answer and Explanation is:
The population of the town after 5 years will be approximately 2092.
Explanation
This problem involves exponential decay, which describes how a quantity decreases at a rate proportional to its current value. To solve this, we use the exponential decay formula, which is a key tool in mathematics for modeling situations like population decline, radioactive decay, or asset depreciation.
The standard formula for exponential decay is:
P(t) = P₀(1 – r)ᵗ
Here is a breakdown of each component of the formula:
- P(t) is the final population after a certain amount of time, t. This is what we need to calculate.
- P₀ is the initial population at the beginning of the time period. In this problem, the initial population is 2500.
- r is the rate of decrease per time period, expressed as a decimal. The problem states the population is decreasing at a rate of 3.5% per year. To convert this percentage to a decimal, we divide by 100, which gives us r = 0.035.
- t is the number of time periods that have passed. In this case, the time period is 5 years.
Now, we can substitute the given values into the formula:
P(5) = 2500(1 – 0.035)⁵
First, we calculate the value inside the parentheses. This term, (1 – r), is known as the decay factor. It represents the proportion of the population that remains from one year to the next.
1 – 0.035 = 0.965
Our equation now becomes:
P(5) = 2500(0.965)⁵
Next, we calculate (0.965)⁵, which means multiplying 0.965 by itself five times.
(0.965)⁵ ≈ 0.8368277
Finally, we multiply this result by the initial population:
P(5) ≈ 2500 * 0.8368277
P(5) ≈ 2092.069
Since population is counted in whole numbers, we round the result to the nearest person. The number 2092.069 rounds down to 2092. Therefore, after 5 years of decreasing at a rate of 3.5% per year, the town’s population is expected to be 2092.
