Question

A home is valued at $236,500 in 2012. In 2017, the home is worth$305,700. Assume the home's value is increasing exponentially. a. Construct a function V(t) which models the value of the home as a function of years since 2012. b. If the model holds, what year will the home value be worth $400,000? Exponential models ANSWERED asked 2020-11-08 A home is valued at$236,500 in 2012. In 2017, the home is worth $305,700. Assume the home's value is increasing exponentially. a. Construct a function V(t) which models the value of the home as a function of years since 2012. b. If the model holds, what year will the home value be worth$400,000?

2020-11-09
Consider the following exponential function of value of the home:
$$\displaystyle{V}{\left({t}\right)}={a}{b}^{{t}}$$
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 $$\displaystyle{a}{b}^{{0}}={236500}$$ $$\displaystyle{a}={236500}$$ In 2017, value of home is$305,700
$$\displaystyle{t}={2017}-{2012}$$
$$\displaystyle={5}$$
$$\displaystyle{236500}\times{b}^{{5}}={305700}$$
$$\displaystyle{b}^{{5}}={\frac{{{305700}}}{{{236500}}}}$$
$$\displaystyle{b}^{{5}}={1.2926}$$
$$\displaystyle{b}=\sqrt{{{5}}}{\left\lbrace{1.2926}\right\rbrace}$$
$$\displaystyle{b}={1.05267}$$
Hence, the exponential function is $$\displaystyle{V}{\left({t}\right)}={236500}\times{\left({1.05267}\right)}^{{t}}={400000}$$
$$\displaystyle{\left({1.05267}\right)}^{{t}}={\frac{{{400000}}}{{{236500}}}}$$
$$\displaystyle{\left({1.05267}\right)}^{{t}}={1.691332}$$
Take logarithm on both sides of the equation:
$$\displaystyle{t}\times{\log{{\left({1.05267}\right)}}}={\log{{\left({1.691332}\right)}}}$$
$$\displaystyle{t}={\frac{{{\log{{\left({1.691332}\right)}}}}}{{{\log{{\left({1.05267}\right)}}}}}}$$
$$\displaystyle={\frac{{{0.228229}}}{{{0.02229}}}}$$
$$\displaystyle={10.23}$$
$$\displaystyle\approx{10}$$
Hence, after 10 years value of home will become \$400,000.