Answer:
The half life of the car is 3.98 years.
Step-by-step explanation:
The value of the car after t years is given by the following equation:
[tex]V(t) = V(0)(1-r)^{t}[/tex]
In which V(0) is the initial value and r is the constant decay rate, as a decimal.
The value of a certain car decreases by 16% each year.
This means that [tex]r = 0.16[/tex]
So
[tex]V(t) = V(0)(1-r)^{t}[/tex]
[tex]V(t) = V(0)(1-0.16)^{t}[/tex]
[tex]V(t) = V(0)(0.84)^{t}[/tex]
What is the 1⁄2-life of the car?
This is t for which V(t) = 0.5V(0). So
[tex]V(t) = V(0)(0.84)^{t}[/tex]
[tex]0.5V(0) = V(0)(0.84)^{t}[/tex]
[tex](0.84)^{t} = 0.5[/tex]
[tex]\log{(0.84)^{t}} = \log{0.5}[/tex]
[tex]t\log{0.84} = \log{0.5}[/tex]
[tex]t = \frac{\log{0.5}}{\log{0.84}}[/tex]
[tex]t = 3.98[/tex]
The half life of the car is 3.98 years.
