A new car is purchased for 17000 dollars. The value of the car depreciates at $$ 13.5 \% $$ per year. What will the value of the car be, to the nearest cent, after 12 years?

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To find the value of the car after 12 years, we can use the formula for depreciation:

$$ V = P \times (1 - r)^n $$

where:
- $$ V $$ is the value of the car after $$ n $$ years,
- $$ P $$ is the initial value of the car,
- $$ r $$ is the annual depreciation rate,
- $$ n $$ is the number of years.

Given:
- Initial value of the car, $$ P = 17000 $$ dollars,
- Annual depreciation rate, $$ r = 13.5\% = 0.135 $$,
- Number of years, $$ n = 12 $$.

Substitute the values into the formula:

$$ V = 17000 \times (1 - 0.135)^{12} $$

Now, we can calculate the value of the car after 12 years.
Calculate the value by following steps:
- step0: Calculate:
  $$17000\left(1-0.135\right)^{12}$$
- step1: Subtract the numbers:
  $$17000\times 0.865^{12}$$
- step2: Convert the expressions:
  $$17000\left(\frac{173}{200}\right)^{12}$$
- step3: Simplify:
  $$17000\times \frac{173^{12}}{200^{12}}$$
- step4: Rewrite the expression:
  $$200\times 85\times \frac{173^{12}}{200^{12}}$$
- step5: Reduce the numbers:
  $$85\times \frac{173^{12}}{200^{11}}$$
- step6: Rewrite the expression:
  $$5\times 17\times \frac{173^{12}}{200^{11}}$$
- step7: Rewrite the expression:
  $$5\times 17\times \frac{173^{12}}{25^{11}\times 8^{11}}$$
- step8: Rewrite the expression:
  $$5\times 17\times \frac{173^{12}}{5^{22}\times 8^{11}}$$
- step9: Reduce the numbers:
  $$17\times \frac{173^{12}}{5^{21}\times 8^{11}}$$
- step10: Multiply the fractions:
  $$\frac{17\times 173^{12}}{5^{21}\times 8^{11}}$$
The value of the car after 12 years will be approximately $2982.92 to the nearest cent.
The car will be worth approximately $2,982.92 after 12 years.

A new car is purchased for 17000 dollars. The value of the car depreciates at \( 13.5 \% \) per year. What will the value of the car be, to the nearest cent, after 12 years?

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