The population of a rural region is expected to decrease by 2% per year for the next 10 years. If the current population of the region is 100,000 people, what will be the expected population after 10 years, assuming the rate of decrease remains constant and compounds annually?

Introduction:
This question applies the compound interest concept to population decrease instead of financial growth. The population shrinks by a fixed percentage each year, so we use a decay factor rather than a growth factor to predict the population after a given number of years.

Given Data / Assumptions:

• Initial population P0 = 100,000 people.
• Annual rate of decrease = 2% per year.
• Time period = 10 years.
• The percentage change applies in a compounded manner each year.

Concept / Approach:
A yearly decrease of 2% means that each year the population is 98% (100% - 2%) of the previous year. The decay factor is therefore 0.98. The population after t years is given by:
Pt = P0 * (1 - r)^twhere r is the decimal decrease rate. Here r = 0.02 and t = 10.

Step-by-Step Solution:
Step 1: Convert the percentage decrease to a decimal: r = 2% = 0.02.Step 2: Compute the decay factor: 1 - r = 1 - 0.02 = 0.98.Step 3: Express the population after 10 years as Pt = 100,000 * (0.98)^10.Step 4: Compute (0.98)^10 approximately. We know (0.98)^2 ≈ 0.9604 and repeated multiplication leads to about 0.8171 after 10 factors.Step 5: Multiply by the initial population: Pt ≈ 100,000 * 0.8171 = 81,710 approximately.Step 6: Rounding to the nearest whole number in the options, we get about 81,707.Thus the expected population after 10 years is close to 81,707 people.

Verification / Alternative check:
We can check quickly by noting that 2% of 100,000 is 2,000. A linear (non compounded) decrease would suggest a 20,000 drop over 10 years, giving about 80,000. Because the decrease is compounded, the annual reduction shrinks over time, leading to a final value slightly above 80,000, which aligns well with 81,707 as given in the answer choices.

Why Other Options Are Wrong:
91,707 assumes growth rather than decline or a much smaller effective decrease. 71,707 and 61,707 imply much faster shrinkage than 2% per year, and 70,000 aligns more with a simple 3% or greater decline. The only value that matches a 2% compounded decrease over 10 years from 100,000 is approximately 81,707.

Common Pitfalls:
Students sometimes treat percentage decrease linearly and multiply 2% by 10, then subtract 20% of the population in one go, giving 80,000. This ignores the compounding nature of the change. Another issue is using 1.02 instead of 0.98, which would represent growth, not decline. Always adjust the base by (1 - r) for decreases and raise this factor to the power of the number of years.

Final Answer:
The expected population of the region after 10 years, given a 2% annual compounded decrease, is approximately 81,707 people.