Answer:
The rate of decay each month is -0.006924%
Step-by-step explanation:
The cost of a car after 1 year at a 8% decreased
Decreased price = Sale price - (Sale price x porcentage of decreased/100%)
Decreased price = $20,000 - ($20,000 x 8%/100%)
Decreased price = $20,000 - ($20,000 x 0.08)
Decreased price = $20,000 - $1,600 = $18,400
So, the value of the car decreased by 8% in a year is $18,400.
The general equation for exponential decay is:
![y=C(1-r)^{t}](https://tex.z-dn.net/?f=y%3DC%281-r%29%5E%7Bt%7D)
Where
y = final amount
C = Sale price
r = rate of decay
t = time
We know that the sale price was $20,000. After a year the decreased price is $18,400.
From the general equation for exponential decay, our sale price is $20,000 decreased over a time of 12 months resulting the final amount of $18,400.
substituting the values
![18,400=20,000(1-r)^{12}](https://tex.z-dn.net/?f=18%2C400%3D20%2C000%281-r%29%5E%7B12%7D)
Solving the equation for r
![\frac{18,400}{20,000}=\frac{20,000}{20,000}(1-r)^{12}](https://tex.z-dn.net/?f=%5Cfrac%7B18%2C400%7D%7B20%2C000%7D%3D%5Cfrac%7B20%2C000%7D%7B20%2C000%7D%281-r%29%5E%7B12%7D)
![\frac{18,400}{20,000}=(1-r)^{12}](https://tex.z-dn.net/?f=%5Cfrac%7B18%2C400%7D%7B20%2C000%7D%3D%281-r%29%5E%7B12%7D)
![\frac{18,400}{20,000}^{\frac{1}{12} } =[(1-r)^{12}]^{\frac{1}{12}}](https://tex.z-dn.net/?f=%5Cfrac%7B18%2C400%7D%7B20%2C000%7D%5E%7B%5Cfrac%7B1%7D%7B12%7D%20%7D%20%3D%5B%281-r%29%5E%7B12%7D%5D%5E%7B%5Cfrac%7B1%7D%7B12%7D%7D)
![\frac{18,400}{20,000}^{\frac{1}{12} } =1-r](https://tex.z-dn.net/?f=%5Cfrac%7B18%2C400%7D%7B20%2C000%7D%5E%7B%5Cfrac%7B1%7D%7B12%7D%20%7D%20%3D1-r)
![r+\frac{18,400}{20,000}^{\frac{1}{12} } =1-r+r](https://tex.z-dn.net/?f=r%2B%5Cfrac%7B18%2C400%7D%7B20%2C000%7D%5E%7B%5Cfrac%7B1%7D%7B12%7D%20%7D%20%3D1-r%2Br)
![r+\frac{18,400}{20,000}^{\frac{1}{12} } =1](https://tex.z-dn.net/?f=r%2B%5Cfrac%7B18%2C400%7D%7B20%2C000%7D%5E%7B%5Cfrac%7B1%7D%7B12%7D%20%7D%20%3D1)
![r+\frac{18,400}{20,000}^{\frac{1}{12} }-\frac{18,400}{20,000}^{\frac{1}{12} } =1-\frac{18,400}{20,000}^{\frac{1}{12} }](https://tex.z-dn.net/?f=r%2B%5Cfrac%7B18%2C400%7D%7B20%2C000%7D%5E%7B%5Cfrac%7B1%7D%7B12%7D%20%7D-%5Cfrac%7B18%2C400%7D%7B20%2C000%7D%5E%7B%5Cfrac%7B1%7D%7B12%7D%20%7D%20%3D1-%5Cfrac%7B18%2C400%7D%7B20%2C000%7D%5E%7B%5Cfrac%7B1%7D%7B12%7D%20%7D)
![r=1-\frac{18,400}{20,000}^{\frac{1}{12}}](https://tex.z-dn.net/?f=r%3D1-%5Cfrac%7B18%2C400%7D%7B20%2C000%7D%5E%7B%5Cfrac%7B1%7D%7B12%7D%7D)
![r=1-\sqrt[12]{\frac{18,400}{20,000}}](https://tex.z-dn.net/?f=r%3D1-%5Csqrt%5B12%5D%7B%5Cfrac%7B18%2C400%7D%7B20%2C000%7D%7D)
![r=1-\sqrt[12]{0.92}\\r=1-0.993076\\r= -0.006924](https://tex.z-dn.net/?f=r%3D1-%5Csqrt%5B12%5D%7B0.92%7D%5C%5Cr%3D1-0.993076%5C%5Cr%3D%20-0.006924)