According to the 2010 United States Census the population of

According to the 2010 United States Census, the population of a town was 98682, roughly a 28% increase from 77095 in 2000.

Construct an exponential model to predict the town\'s population each decade. Use your model to predict the population of the town in 2050.
(Express your answer rounded up to the nearest person.)

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Solution

Let an exponential model to predict the town\'s population each decade be y = f(x) = ab^x, where a,b are arbitrary real numbers and f(x) is the town\'s population x decades from 2000. Since the population of the town was 77095 in 2000, on substituting x = 0, and y = f(0) = 77095,we get 77095 = ab⁰ or, a = 77095 ( as b⁰ =1). Hence, the required exponential model changes to y = f(x) = 77095b^x. Further, since the population of the town was 98682 in 2010, on substituting x = 1, and y = f(1) = 98682,we get 98682 =77095b¹ or, b = 98682/77095 =1.280005188 = 1.28 (say, on rounding off to 2 decimal places). Then, we have y = f(x) = 77095(98682/77095 )^x= 77095 (1.28)^x. However, if we do not round off at this stage, we will get a more accurate result for future predictions.

Now, when x = 5, the population of the town in 2050 is likely to be f(5) = 77095 (98682/77095)⁵ = 77095*(1.280005188)⁵ = 77095*3.436043475 = 264902 ( on rounding off to the nearest whole number).

Note:

If we round off to 1.28 earlier, then, the result would be 264896.