SOLUTION: 2. In 2010, the population of a town is 8500. The population decreases by 4.5% each year. (a) Write an equation to find the population of the town t years after 2010. (b) In what

Algebra ->  Equations -> SOLUTION: 2. In 2010, the population of a town is 8500. The population decreases by 4.5% each year. (a) Write an equation to find the population of the town t years after 2010. (b) In what      Log On


   

Question 1127544: 2. In 2010, the population of a town is 8500. The population decreases by 4.5% each year.
(a) Write an equation to find the population of the town t years after 2010.
(b) In what year will the population of the town be 7000? Show your work.
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


If the population decreases by 4.5% per year, then each year the population is 95.5% of what it was the year before. So the current population gets multiplied by 0.955 to get the population in the next year. With the population of 8500 in 2010, the population t years after 2010 will be

8500%2A%280.955%29%5Et

To find when the population will be 7000, solve the equation

8500%2A%280.955%29%5Et=7000

The easiest way to get the answer is with a graphing calculator, which shows the solution to be 4.22 to 2 decimal places, which means it will be during 2014 that the population will be 7000.

For an algebraic solution, with the variable in the exponent, you need to use logarithms.

8500%2A%280.955%29%5Et=7000
0.955%5Et+=+7000%2F8500+=+14%2F17
t%2Alog%28%280.955%29%29+=+log%28%2814%2F17%29%29
t+=+log%28%2814%2F17%29%29%2Flog%28%280.955%29%29

which, to 2 decimal places, is 4.22.

ANSWER: The population will reach 7000 in the year 2014.