2020-11-08

A home is valued at $236,500 in 2012. In 2017, the home is worth$305,700. Assume the homes

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Consider the following exponential function of value of the home:
$V\left(t\right)=a{b}^{t}$
Where,
V(t)=Value of home after t years
t=number of years from 2012
a and b are constants
In 2012, value of home is $236,500 $a{b}^{0}=236500$ $a=236500$ In 2017, value of home is$305,700
$t=2017-2012$
$=5$
$236500×{b}^{5}=305700$
${b}^{5}=\frac{305700}{236500}$
${b}^{5}=1.2926$
$b=\sqrt{5}\left\{1.2926\right\}$
$b=1.05267$
Hence, the exponential function is $V\left(t\right)=236500×{\left(1.05267\right)}^{t}=400000$
${\left(1.05267\right)}^{t}=\frac{400000}{236500}$
${\left(1.05267\right)}^{t}=1.691332$
Take logarithm on both sides of the equation:
$t×\mathrm{log}\left(1.05267\right)=\mathrm{log}\left(1.691332\right)$
$t=\frac{\mathrm{log}\left(1.691332\right)}{\mathrm{log}\left(1.05267\right)}$
$=\frac{0.228229}{0.02229}$
$=10.23$
$\approx 10$
Hence, after 10 years value of home will become \$400,000.

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