Let's solve the problem step by step.
The dollar value [tex]\( v(t) \)[/tex] of the car as a function of its age [tex]\( t \)[/tex] in years is given by the formula:
[tex]\[
v(t) = 25,900 \cdot (0.92)^t
\][/tex]
### Step 1: Initial Value of the Car
The initial value of the car is the value when [tex]\( t = 0 \)[/tex]. At [tex]\( t = 0 \)[/tex]:
[tex]\[
v(0) = 25,900 \cdot (0.92)^0
\][/tex]
Since any number raised to the power of 0 is 1:
[tex]\[
v(0) = 25,900 \cdot 1 = 25,900
\][/tex]
So, the initial value of the car is:
[tex]\[
\$25,900
\][/tex]
### Step 2: Value of the Car After 10 Years
To find the value of the car after 10 years, we plug [tex]\( t = 10 \)[/tex] into the formula:
[tex]\[
v(10) = 25,900 \cdot (0.92)^{10}
\][/tex]
After performing the calculation:
[tex]\[
v(10) \approx 11,251
\][/tex]
So, the value of the car after 10 years is:
[tex]\[
\$\boxed{11,251}
\][/tex]
### Summary
- Initial value: \[tex]$25,900
- Value after 10 years: \$[/tex]11,251
