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
[tex]y=ab^x[/tex] ...(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).
[tex]y=95000b^x[/tex] ...(ii)
The value of home in 2018 is $105,000. It means the value of y is 105000 at x=7.
[tex]105000=95000b^7[/tex]
[tex]\dfrac{105000}{95000}=b^7[/tex]
[tex]\dfrac{21}{19}=b^7[/tex]
Taking 7th root on both sides, we get
[tex]\left(\dfrac{21}{19}\right)^{\frac{1}{7}}=b[/tex]
Put [tex]b=\left(\dfrac{21}{19}\right)^{\frac{1}{7}}[/tex] in (ii).
[tex]y=95000\left(\left(\dfrac{21}{19}\right)^{\frac{1}{7}}\right)^x[/tex]
[tex]y=95000\left(\dfrac{21}{19}\right)^{\frac{x}{7}}[/tex]
Therefore, the required exponential model for the value of home is [tex]y=95000\left(\dfrac{21}{19}\right)^{\frac{x}{7}}[/tex], where x is the number of years after 2011.
