Question

A young couple purchases their first new home in 2011 for​ $95,000. They sell it to move into a bigger home in 2018 for​ $105,000. First, we will develop an exponential model for the value of the home. The model will have the form

Answer 1

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

$y=ab^x$       ...(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).

$y=95000b^x$       ...(ii)

The value of home in 2018 is $105,000. It means the value of y is 105000 at x=7.

$105000=95000b^7$

$\dfrac{105000}{95000}=b^7$

$\dfrac{21}{19}=b^7$

Taking 7th root on both sides, we get

$\left(\dfrac{21}{19}\right)^{\frac{1}{7}}=b$

Put $b=\left(\dfrac{21}{19}\right)^{\frac{1}{7}}$ in (ii).

$y=95000\left(\left(\dfrac{21}{19}\right)^{\frac{1}{7}}\right)^x$

$y=95000\left(\dfrac{21}{19}\right)^{\frac{x}{7}}$

Therefore, the required exponential model for the value of home is $y=95000\left(\dfrac{21}{19}\right)^{\frac{x}{7}}$, where x is the number of years after 2011.