Population projection — incremental increase method: A city had a population of 50,000 in the year 2000. Over the last 8 decades, the average increase per decade has been 7,500 and the average incremental increase per decade has been 750. Using the incremental increase method, estimate the population in 2020.

Introduction / Context:
Population forecasting guides water-demand estimation and infrastructure sizing. The incremental increase method refines the simple arithmetical method by considering not only the average decadal increase but also the trend in that increase (i.e., the incremental rise of the decadal growth itself).

Given Data / Assumptions:

• Base population in 2000 (P0) = 50,000.
• Average decadal increase (A) = 7,500.
• Average incremental increase per decade (I) = 750.
• Projection horizon: 2020 → n = 2 decades ahead.

Concept / Approach:
In the incremental increase method, the total added population over n decades is taken as n*A plus an allowance for the trend of increasing increment. A commonly used planner’s shortcut for two decades is to add A for the first decade and (A + I) for the second decade. This captures the baseline increase and one step of incremental rise.

Step-by-Step Solution:

Verification / Alternative check:
Some texts express the allowance via compact formulas. Small rounding or convention differences can yield values within a narrow band; selection among given options aligns with the nearest conventional planning figure.

Why Other Options Are Wrong:
55,000 and 60,500 ignore the strong rising trend; 72,500 and 76,500 grossly overstate growth for the given averages.

Common Pitfalls:
Applying geometric growth when data indicate linear-plus-increment behavior; omitting the incremental allowance for the second decade.

Final Answer:
66,500