Question

The function f(x)=501170〖(0.98)〗^x gives the population of a Texas city x years after 1995.

What was the population in 1995?

Is the population increasing or decreasing?

What is the rate of population change?

Based on the function, what is the population predicted to be in the year 2020?

Answer 1

The population in 1995

501170

Is the population increasing or decreasing?

The population is decreasing

The rate of population change

`-10124.99(0.98^(x) )`

population in the year 2020

302439

Explanation:

Since 1995 is the base year, the population in 1995 is simply found by setting x equal to 0 in the given function;

`f(0)=501170(0.98^(0))=501170`

The population is clearly decreasing over time since the base of the exponent, 0.98, is less than 1.

The rate of population change is obtained by differentiating the function with respect to x using the rule;

`if y=a^(x)\\(dy)/(dx)=a^(x)*lna\\`

The population in the year 2020 is obtained by setting x equal to 25 in the population function since 2020-1995 = 25;

`f(25)=501170(0.98^(25)) =302439`