Given:
The value of home in 2011 is $95,000.
The value of home in 2018 is $105,000.
To find:
The exponential model for the value of the home.
Solution:
The general exponential model is
...(i)
where, a is initial value and b is growth factor.
Let 2011 is initial year and x be the number of years after 2011.
So, initial value of home is 95,000, i.e., a=95,000.
Put a=95000 in (i).
...(ii)
The value of home in 2018 is $105,000. It means the value of y is 105000 at x=7.
![105000=95000b^7](https://tex.z-dn.net/?f=105000%3D95000b%5E7)
![\dfrac{21}{19}=b^7](https://tex.z-dn.net/?f=%5Cdfrac%7B21%7D%7B19%7D%3Db%5E7)
Taking 7th root on both sides, we get
![\left(\dfrac{21}{19}\right)^{\frac{1}{7}}=b](https://tex.z-dn.net/?f=%5Cleft%28%5Cdfrac%7B21%7D%7B19%7D%5Cright%29%5E%7B%5Cfrac%7B1%7D%7B7%7D%7D%3Db)
Put
in (ii).
![y=95000\left(\left(\dfrac{21}{19}\right)^{\frac{1}{7}}\right)^x](https://tex.z-dn.net/?f=y%3D95000%5Cleft%28%5Cleft%28%5Cdfrac%7B21%7D%7B19%7D%5Cright%29%5E%7B%5Cfrac%7B1%7D%7B7%7D%7D%5Cright%29%5Ex)
![y=95000\left(\dfrac{21}{19}\right)^{\frac{x}{7}}](https://tex.z-dn.net/?f=y%3D95000%5Cleft%28%5Cdfrac%7B21%7D%7B19%7D%5Cright%29%5E%7B%5Cfrac%7Bx%7D%7B7%7D%7D)
Therefore, the required exponential model for the value of home is
, where x is the number of years after 2011.