  Question

In planning maintenance for a city's infrastructure, a civil engineer estimates that, starting from the present, the population of the city will decrease by $$10$$ percent every $$20$$ years. If the present population of the city is $$50,000$$, which of the following expressions represents the engineers estimate of the population of the city $$t$$ years from now?

A
50,000(0.1)2t  B
50,000(0.1)t2  C
50,000(0.9)2t  D
50,000(0.9)t20  Solution

The correct option is D $$50,000(0.9)^{\tfrac {t}{20}}$$For the present population to decrease by $$10$$ percent, it must be multiplied by the factor $$\left(1-\dfrac{10}{100}\right) =(1-0.1) =0.9$$. Since the engineer estimates that the population will decrease by $$10$$ percent every $$20$$ years, the present population, $$50,000$$, must be multiplied by $$(0.9)^{ n }$$, where $$n$$ is the number of $$20$$‑year periods that will have elapsed $$t$$ years from now. After $$t$$ years, the number of $$20$$‑year periods that have elapsed is $$\dfrac { t }{ 20 }$$. Therefore, $$50,000(0.9)^{ \tfrac { t }{ 20 } }$$ represents the engineer’s estimate of the population of the city $$t$$ years from now.Mathematics

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