To determine how old the car is, we start by using the information provided:
- [tex]\( y \)[/tex] is the current value, which is half of the original cost, so [tex]\( y = \frac{A}{2} \)[/tex].
- [tex]\( A \)[/tex] is the original cost of the car.
- [tex]\( r \)[/tex] is the rate of depreciation, which is given as 10%, or [tex]\( r = 0.10 \)[/tex].
- [tex]\( t \)[/tex] is the time in years, which we need to find.
We are given the equation for depreciation:
[tex]\[ y = A(1-r)^t \][/tex]
Substituting the known values into the equation:
[tex]\[ \frac{A}{2} = A(1 - 0.10)^t \][/tex]
Simplifying this equation, we divide both sides by [tex]\( A \)[/tex]:
[tex]\[ \frac{1}{2} = (0.90)^t \][/tex]
To solve for [tex]\( t \)[/tex], we need to apply logarithms. Taking the natural logarithm on both sides:
[tex]\[ \ln\left(\frac{1}{2}\right) = \ln\left((0.90)^t\right) \][/tex]
Using the properties of logarithms, particularly [tex]\(\ln(a^b) = b \ln(a)\)[/tex], we get:
[tex]\[ \ln\left(\frac{1}{2}\right) = t \ln(0.90) \][/tex]
Solving for [tex]\( t \)[/tex] by isolating it:
[tex]\[ t = \frac{\ln\left(\frac{1}{2}\right)}{\ln(0.90)} \][/tex]
Now we calculate the values of the logarithms:
[tex]\[ \ln\left(\frac{1}{2}\right) \approx -0.6931 \][/tex]
[tex]\[ \ln(0.90) \approx -0.1054 \][/tex]
Dividing these values:
[tex]\[ t \approx \frac{-0.6931}{-0.1054} \approx 6.578813478960585 \][/tex]
Thus, the approximate age of the car is around 6.58 years.
Given the multiple-choice options:
- 3.3 years
- 5.0 years
- 5.6 years
- 6.6 years
The closest value to 6.58 is 6.6 years. Therefore, the car is approximately 6.6 years old.
