Activity 3.1.3.
The value of an automobile is depreciating. When the car is \(3\) years old, its value is $\(12500\text{;}\) when the car is \(7\) years old, its value is $\(6500\text{.}\)
(a)
Suppose the car’s value \(t\) years after its purchase is given by the function \(V(t)\) and that \(V\) is exponential with form \(V(t) = ab^t\text{,}\) what are the values of \(a\) and \(b\text{?}\) Find \(a\) and \(b\) both exactly and approximately.
(b)
Using the exponential model determined in (a), determine the purchase value of the car and then use Desmos to estimate when the car will be worth less than $\(1000\text{.}\)
(c)
Suppose instead that the car’s value is modeled by a linear function \(L\) and satisfies the values stated at the outset of this activity. Find a formula for \(L(t)\) and determine both the purchase value of the car and when the car will be worth $\(1000\text{.}\)
(d)
Which model do you think is more realistic? Why?
