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:
[tex]y=C(1-r)^{t}[/tex]
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
[tex]18,400=20,000(1-r)^{12}[/tex]
Solving the equation for r
[tex]\frac{18,400}{20,000}=\frac{20,000}{20,000}(1-r)^{12}[/tex]
[tex]\frac{18,400}{20,000}=(1-r)^{12}[/tex]
[tex]\frac{18,400}{20,000}^{\frac{1}{12} } =[(1-r)^{12}]^{\frac{1}{12}}[/tex]
[tex]\frac{18,400}{20,000}^{\frac{1}{12} } =1-r[/tex]
[tex]r+\frac{18,400}{20,000}^{\frac{1}{12} } =1-r+r[/tex]
[tex]r+\frac{18,400}{20,000}^{\frac{1}{12} } =1[/tex]
[tex]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} }[/tex]
[tex]r=1-\frac{18,400}{20,000}^{\frac{1}{12}}[/tex]
[tex]r=1-\sqrt[12]{\frac{18,400}{20,000}}[/tex]
[tex]r=1-\sqrt[12]{0.92}\\r=1-0.993076\\r= -0.006924[/tex]
