Answer:
(a)Initial Value: $29,900
(b)Value after 12 years: $1516.37
Explanation:
Given the dollar value v(t) of a certain car model that is t years old as:
[tex]v\mleft(t\mright)=29,900\mleft(0.78\mright)^t[/tex](a)Initial Value
At initial value, that is, when the car was just purchased, t=0
[tex]\begin{gathered} v\mleft(t\mright)=29,900\mleft(0.78\mright)^t \\ \implies v\mleft(0\mright)=29,900\mleft(0.78\mright)^0=29,900\times1=29,900 \end{gathered}[/tex]The initial value of the car is $29,900.
(b)The value after 12 years.
When t=12
[tex]\begin{gathered} v\mleft(t\mright)=29,900\mleft(0.78\mright)^t \\ \implies v\mleft(12\mright)=29,900\mleft(0.78\mright)^{12}=\$1516.37 \end{gathered}[/tex]The value of the car after 12 years is $1516.37.