Answer:
a. ![y=830*(0.87)^x](https://tex.z-dn.net/?f=y%3D830%2A%280.87%29%5Ex)
b. The value of stereo system after 2 years will be $628.23.
c. After approximately 4.98 years the stereo will be worth half the original value.
Step-by-step explanation:
Let x be the number of years.
We have been given that you purchased a stereo system for $830. The value of the stereo system decreases 13% each year.
a. Since we know that an exponential function is in form:
, where,
a = Initial value,
b = For decay b is in form (1-r), where r is rate in decimal form.
Let us convert our given rate in decimal form.
![13\%=\frac{13}{100}=0.13](https://tex.z-dn.net/?f=13%5C%25%3D%5Cfrac%7B13%7D%7B100%7D%3D0.13)
Upon substituting our given values in exponential decay function we will get
![y=830*(1-0.13)^x](https://tex.z-dn.net/?f=y%3D830%2A%281-0.13%29%5Ex)
Therefore, the exponential model
represents the value of the stereo system in terms of the number of years since the purchase.
b. To find the value of stereo system after 2 years we will substitute x=2 in our model.
![y=830*(0.87)^2](https://tex.z-dn.net/?f=y%3D830%2A%280.87%29%5E2)
![y=830*0.7569](https://tex.z-dn.net/?f=y%3D830%2A0.7569)
![y=628.227\approx 628.23](https://tex.z-dn.net/?f=y%3D628.227%5Capprox%20628.23)
Therefore, the value of stereo system after 2 years will be $628.23.
c. The half of the original price will be
.
Let us substitute y=415 in our model to find the time it will take the stereo to be worth half the original value.
![415=830*(0.87)^x](https://tex.z-dn.net/?f=415%3D830%2A%280.87%29%5Ex)
Upon dividing both sides of our equation by 830 we will get,
![\frac{415}{830}=\frac{830*(0.87)^x}{830}](https://tex.z-dn.net/?f=%5Cfrac%7B415%7D%7B830%7D%3D%5Cfrac%7B830%2A%280.87%29%5Ex%7D%7B830%7D)
![0.5=0.87^x](https://tex.z-dn.net/?f=0.5%3D0.87%5Ex)
Let us take natural log of both sides of our equation.
![ln(0.5)=ln(0.87^x)](https://tex.z-dn.net/?f=ln%280.5%29%3Dln%280.87%5Ex%29)
Using natural log property
we will get,
![ln(0.5)=x*ln(0.87)](https://tex.z-dn.net/?f=ln%280.5%29%3Dx%2Aln%280.87%29)
![\frac{ln(0.5)}{ln(0.87)}=\frac{x*ln(0.87)}{ln(0.87)}](https://tex.z-dn.net/?f=%5Cfrac%7Bln%280.5%29%7D%7Bln%280.87%29%7D%3D%5Cfrac%7Bx%2Aln%280.87%29%7D%7Bln%280.87%29%7D)
![\frac{ln(0.5)}{ln(0.87)}=x](https://tex.z-dn.net/?f=%5Cfrac%7Bln%280.5%29%7D%7Bln%280.87%29%7D%3Dx)
![\frac{-0.6931471805599}{-0.139262067}=x](https://tex.z-dn.net/?f=%5Cfrac%7B-0.6931471805599%7D%7B-0.139262067%7D%3Dx)
![x=4.977286\approx 4.98](https://tex.z-dn.net/?f=x%3D4.977286%5Capprox%204.98)
Therefore, after approximately 4.98 years the stereo will be worth half the original value.