A new car is purchased for 17000 dollars. The value of the car depreciates
at per year. What will the value of the car be, to the nearest cent,
after 14 years?

Question

A new car is purchased for 17000 dollars. The value of the car depreciates at $$ 12.25 \% $$ per year. What will the value of the car be, to the nearest cent, after 14 years?

UpStudy AI · Accepted Answer

To find the value of the car after 14 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, $$ r = 12.25\% $$,
- Number of years, $$ n = 14 $$.

Substitute the given values into the formula:

$$ V = 17000 \times (1 - 0.1225)^{14} $$

Now, we can calculate the value of the car after 14 years.
Calculate the value by following steps:
- step0: Calculate:
  $$17000\left(1-0.1225\right)^{14}$$
- step1: Subtract the numbers:
  $$17000\times 0.8775^{14}$$
- step2: Convert the expressions:
  $$17000\left(\frac{351}{400}\right)^{14}$$
- step3: Simplify:
  $$17000\times \frac{351^{14}}{400^{14}}$$
- step4: Rewrite the expression:
  $$200\times 85\times \frac{351^{14}}{400^{14}}$$
- step5: Rewrite the expression:
  $$200\times 85\times \frac{351^{14}}{200^{14}\times 2^{14}}$$
- step6: Reduce the numbers:
  $$85\times \frac{351^{14}}{200^{13}\times 2^{14}}$$
- step7: Rewrite the expression:
  $$5\times 17\times \frac{351^{14}}{200^{13}\times 2^{14}}$$
- step8: Rewrite the expression:
  $$5\times 17\times \frac{351^{14}}{25^{13}\times 8^{13}\times 2^{14}}$$
- step9: Rewrite the expression:
  $$5\times 17\times \frac{351^{14}}{5^{26}\times 8^{13}\times 2^{14}}$$
- step10: Reduce the numbers:
  $$17\times \frac{351^{14}}{5^{25}\times 8^{13}\times 2^{14}}$$
- step11: Multiply the fractions:
  $$\frac{17\times 351^{14}}{5^{25}\times 2^{53}}$$
The value of the car after 14 years will be approximately $2728.40 to the nearest cent.
The car will be worth approximately $2,728.40 after 14 years.

A new car is purchased for 17000 dollars. The value of the car depreciates
at per year. What will the value of the car be, to the nearest cent,
after 14 years?

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Solución

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A new car is purchased for 17000 dollars. The value of the car depreciates at per year. What will the value of the car be, to the nearest cent, after 14 years?

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A new car is purchased for 17000 dollars. The value of the car depreciates
at per year. What will the value of the car be, to the nearest cent,
after 14 years?