Respuesta :
Given:
Original Value - $29,000
Final Value after 13 years - $15,000
Find: annual rate of change between 1993 and 2006 (13 years)
Solution:
To determine the annual rate of change, we use the formula below:
[tex]y=ab^x[/tex]where a = the initial value, b = the rate of change, x = the number of years, and y = the final value at x year.
Since we already have the values of the variables provided in the question, let's replace them.
[tex]15,000=29,000(b)^{13}[/tex]Then, solve for b.
Divide both sides of the equation by 29,000.
[tex]\frac{15,000}{29,000}=\frac{29,000b^{13}}{29,000}\Rightarrow\frac{15}{29}=b^{13}[/tex]Power both sides of the equation by 1/13.
[tex](\frac{15}{29})^{\frac{1}{13}}=(b^{13})^{\frac{1}{13}}\Rightarrow0.95055[/tex]The value of b is 0.95055.
Let's subtract this from 1.
[tex]1-0.95055=0.04945[/tex]Multiply this by 100.
[tex]0.04945\times100=4.945\approx4.95\%[/tex]Therefore, the annual rate of change between 1993 to 2006 is -4.95%. We used negative because we are dealing with depreciation. The depreciation rate annually is 4.95%.
If the value of the car continues to drop after an additional 3 years (2006 - 2009) or 16 years from 1993, let's replace "x" in the equation above with 16, and let's use b = 0.95055 to solve for the value of the card in the year 2009.
[tex]y_{2009}=29,000(0.95055)^{16}[/tex]Type those values in a calculator and solve.
[tex]y_{2009}\approx12,882.42[/tex]The value of the car in the year 2009 will be approximately $12, 882.42.
