Respuesta :
Using an exponential function, it is found that it would take 19.68 years for the value of the car to be $3,800.
What is an exponential function?
A decaying exponential function is modeled by:
[tex]A(t) = A(0)(1 - r)^\frac{t}{n}[/tex]
In which:
- A(0) is the initial value.
- r is the decay rate, as a decimal, after each n years.
In this problem:
- The initial value is of $31,000, hence A(0) = 31000.
- It depreciates by one-half every 6.5 years, hence r = 0.5, n = 6.5.
Then, the equation is:
[tex]A(t) = A(0)(1 - r)^\frac{t}{n}[/tex]
[tex]A(t) = 31000(1 - 0.5)^\frac{t}{6.5}[/tex]
[tex]A(t) = 31000(0.5)^\frac{t}{6.5}[/tex]
The value would be of $3,800 at t for which A(t) = 3800, hence:
[tex]A(t) = 31000(0.5)^\frac{t}{6.5}[/tex]
[tex]3800 = 31000(0.5)^\frac{t}{6.5}[/tex]
[tex](0.5)^\frac{t}{6.5} = \frac{3800}{31000}[/tex]
[tex](0.5)^\frac{t}{6.5} = 0.12258064516[/tex]
[tex]\log{(0.5)^\frac{t}{6.5}} = \log{0.12258064516}[/tex]
[tex]\frac{t}{6.5}\log{0.5} = \log{0.12258064516}[/tex]
[tex]t = 6.5\frac{\log{0.12258064516}}{\log{0.5}}[/tex]
t = 19.68.
It would take 19.68 years for the value of the car to be $3,800.
More can be learned about exponential functions at https://brainly.com/question/25537936
