Answer:
In 2026 car will have a value of $5,000.
Step-by-step explanation:
We have been given that Jack bought a new car in 2014 for 28,000. If the value of the car decreases by 14% each year.
Since we know that an exponential function is in form:
, where,
a = Initial value,
b = For decay or decrease b is in form (1-r), where r represents decay rate in decimal form.
Let us convert our given decay rate in decimal form.
![14\%=\frac{14}{100}=0.14](https://tex.z-dn.net/?f=14%5C%25%3D%5Cfrac%7B14%7D%7B100%7D%3D0.14)
Upon substituting a =28,000 and r=0.14 in exponential decay function we will get,
, where x represents number of years after 2014.
Therefore, the function
represents the value of car x years after 2014.
To find the number of years it will take to car have the value of $5,000, we will substitute y=5,000 in our function.
![5,000=28,000(0.86)^x](https://tex.z-dn.net/?f=5%2C000%3D28%2C000%280.86%29%5Ex)
Let us divide both sides of our equation by 28,000.
![\frac{5,000}{28,000}=\frac{28,000(0.86)^x}{28,000}](https://tex.z-dn.net/?f=%5Cfrac%7B5%2C000%7D%7B28%2C000%7D%3D%5Cfrac%7B28%2C000%280.86%29%5Ex%7D%7B28%2C000%7D)
![0.1785714285714286=(0.86)^x](https://tex.z-dn.net/?f=0.1785714285714286%3D%280.86%29%5Ex)
Let us take natural log of both sides of our equation.
![ln(0.1785714285714286)=ln((0.86)^x)](https://tex.z-dn.net/?f=ln%280.1785714285714286%29%3Dln%28%280.86%29%5Ex%29)
Using natural log property
we will get,
![ln(0.1785714285714286)=x*ln(0.86)](https://tex.z-dn.net/?f=ln%280.1785714285714286%29%3Dx%2Aln%280.86%29)
![\frac{ln(0.1785714285714286)}{ln(0.86)}=\frac{x*ln(0.86)}{ln(0.86)}](https://tex.z-dn.net/?f=%5Cfrac%7Bln%280.1785714285714286%29%7D%7Bln%280.86%29%7D%3D%5Cfrac%7Bx%2Aln%280.86%29%7D%7Bln%280.86%29%7D)
![\frac{-1.7227665977411033893}{-0.1508228897345836}=x](https://tex.z-dn.net/?f=%5Cfrac%7B-1.7227665977411033893%7D%7B-0.1508228897345836%7D%3Dx)
As in the 12th year after 2014 car will have a value of $5,000, so we will add 12 to 2014 to find the year.
![2014+12=2026](https://tex.z-dn.net/?f=2014%2B12%3D2026)
Therefore, in 2026 car will have a value of $5,000.