Problem
An SUV is purchased new for $\$ 31,500$. (b) Suppose that the vehicle is depreciated so that it holds only $90 \%$ of its value from the previous year. Write an exponential function of the form $y=V_{0} b^{t}$, where $V_{0}$ is the initial value and $t$ is the number of years after purchase. The model for the value of the vehicle $t$ years after purchase is $y=$
Solution
Step 1 ：The initial value of the SUV is given as \(V_{0} = \$ 31,500\).
Step 2 ：The vehicle is depreciated so that it holds only \(90 \%\) of its value from the previous year. This means the base of the exponential function, \(b\), is \(0.90\).
Step 3 ：Therefore, the exponential function that models the value of the vehicle \(t\) years after purchase is: \[y=V_{0} b^{t} = 31500(0.90)^{t}\]
Step 4 ：Final Answer: The model for the value of the vehicle \(t\) years after purchase is \(\boxed{y=31500(0.90)^{t}}\).
