SOLUTION: I need help for this problem please. Population y of one town x years after the year 2000 is modeled by the equation y=12053*1.0345^x 1. what was the population of the town in th

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: I need help for this problem please. Population y of one town x years after the year 2000 is modeled by the equation y=12053*1.0345^x 1. what was the population of the town in th      Log On


   

Question 1090926: I need help for this problem please.
Population y of one town x years after the year 2000 is modeled by the equation y=12053*1.0345^x
1. what was the population of the town in the year 2000?
2. equation for the population of the town as a continuous model?
3. the population is increasing or not? by what percent?
Thank you.

Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
y=12053%2A1.0345%5Ex where
y= population of the town, and
x= number of years after the year 2000.

1. In the year 2000, x=0
1.0345%5E0=1 , because a non-zero number to the zero power is by definition equal to 1 .
So, in the year 2000,
y=12053%2A1.0345%5E0=y=12053%2A1=highlight%2812053%29 .

2. highlight%28y=12053%2Ae%5E%28ln%281.0345%29%2Ax%29%29 is probably the expected/accepted answer,
although highlight%28y=12053%2Ae%5E%280.033918%2Ax%29%29 is a good approximation,
and highlight%28y=12053%2A1.0345%5Ex%29 is a valid answer.

NOTE:
The teacher may not accept any true answer not fitting what was taught in class. That may include formulas, symbols, and formats that I can an only guess at. If you are asked to "show your work", you may be expected to start by writing a general "formula" for an exponential growth model, such as
P=P%5B0%5D%2Ae%5Ert or y=A%2Ae%5Ekx (Letter symbols may vary).
FURTHER EXPLANATION:
In this problem, the word "model" refers to a function/equation (such as as y=12053%2A1.0345%5Ex ) that fits fairly well the data available, and may be a useful "model" to predict population growth in the near future, or estimate what the population was at a time there is no data for.
The word "continuous" refers to the fact although in real life the x and y values jump from one integer to another, a function/equation used "as a continuous model" applies to any real value of x , yields real values of y , and has a graph that is a curve flowing continuously through an infinite number of points, without any gaps.
With that understanding, I would say that y=12053%2A1.0345%5Ex is one equation for the population of the town as a continuous model, but I believe that is not the expected answer.
For any number, equation, or function, there is a variety of representations limited only by your imagination. An exponential function can be written with any base, but in higher math the irrational number
e=approximately2.71828 is a very popular base.
Using base e logarithms of both sides of the equal sign in y=12053%2A1.0345%5Ex ,
we can work our to an equivalent expression:
y=12053%2A1.0345%5Ex
ln%28y%29=ln%2812053%2A1.0345%5Ex%29
ln%28y%29=ln%2812053%29%2Bln%281.0345%5Ex%29
ln%28y%29=ln%2812053%29%2Bx%2Aln%281.0345%29
e%5Eln%28y%29=e%5E%28ln%2812053%29%2Bx%2Aln%281.0345%29%29
e%5Eln%28y%29=e%5Eln%2812053%29%2Ae%5E%28x%2Aln%281.0345%29%29
y=12053%2Ae%5E%28x%2Aln%281.0345%29%29

3. Of course the population is increasing.
On the year 2000, x=0 and the population is y=12053 .
One year later, the model predicts that for x=1 that number will have multiplied times 1.0345 .
That is 0.0345 times more.
It is 0.0345=3.45%2F100=%223.45%25%22 more in one year.
The population is increasing by highlight%28%223.45%25%22%29 per year.
You may be expected to show by calculating the relative change
from y%280%29=12053 for x=0
to y=12053%2A1.0345 for x=1
as %2812053%2A1.0345-12053%29%2F12053=12053%2A%281.0345-1%29%2F12053=0.0345=%223.45%25%22 ,
or you may be expected to apply a formula such as
r-1 to find the percent increase as a decimal for P=P%5B0%5D%2Ae%5Ert .