Step 1 :The problem is asking for the value of the car after 7 years and 11 years. This can be calculated by substituting the values of 7 and 11 for t in the given function \(v(t)=20000(0.90)^t\).
Step 2 :Substitute 7 for t in the function to find the value of the car after 7 years: \(v(7)=20000(0.90)^7\).
Step 3 :Calculate the value to get \(v(7) = 9566\).
Step 4 :Substitute 11 for t in the function to find the value of the car after 11 years: \(v(11)=20000(0.90)^{11}\).
Step 5 :Calculate the value to get \(v(11) = 6276\).
Step 6 :Final Answer: The value of the car after 7 years is \(\boxed{9566}\) dollars and after 11 years is \(\boxed{6276}\) dollars.