The dollar value $$ v(t) $$ of a certain car model that is $$ t $$ years old is given by the following exponential function. $$ v(t)=27,500(0.92)^{t} $$ Find the initial value of the car and the value after 11 years. Round your answers to the nearest dollar as necessary. Initial value: Value after 11 years:

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**Step 1. Find the initial value of the car**

The function is given by

$$
v(t)=27,500(0.92)^t
$$

When the car is new ($$ t = 0 $$):

$$
v(0)=27,500(0.92)^0 = 27,500(1) = 27,500
$$

Thus, the initial value is $$\$27,500$$.

**Step 2. Find the value after 11 years**

Substitute $$ t = 11 $$ into the function:

$$
v(11)=27,500(0.92)^{11}
$$

First, calculate $$ (0.92)^{11} $$. Using logarithms or a calculator, we find

$$
(0.92)^{11} \approx 0.3993
$$

Then,

$$
v(11) \approx 27,500 \times 0.3993 \approx 10,980.75
$$

Rounding to the nearest dollar, the value after 11 years is approximately $$\$10,981$$.

**Final Answers**

Initial value: $$\$27,500$$  
Value after 11 years: $$\$10,981$$
Initial value: \$27,500 Value after 11 years: \$10,981

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