Suppose the population of a town is 3,400 in 2000. The population decreases at a rate of 2% every 20 years. What will be the projected population in 2040? Round your answer to the nearest whole number.

Question

anonymous · Answer

Probably you know that an exponential function is written in this form:

$\large{f(x) = ar^x}$

Where a is the initial value (in this case 3,400) and r is the rate. If the rate were applied every year and you were told to find the value/quantity after the 1st year, you'll set x=1 (the exponent will be 1), right? If it were for the 2nd year, you'll set x = 2 (the exponent will be 2) and so on. 

In this problem, the key is in how to make the exponent 1 when x = 20; 2 when x= 40 and so on (because the rate is applied every 20 years). Well, you can do so, by setting a fraction in the exponent:

$\large{f(x) = ar^{\frac{x}{t}}}$, where t is the time by which the rate applies ''completely'', so to speak.

In the problem it would be in this way:

$\large{f(x) = ar^{\frac{x}{20}}}$

And your r should be the portion of the population that were left after x year.