Question

a new car is purchased for 20700 dollars. the value of the car depreciates at 11% per year. what will the value of the car be, to the nearest cent, after 11 years?

Explanation:

Step1: Recall the depreciation formula

The formula for exponential depreciation is \( V = P(1 - r)^t \), where \( V \) is the final value, \( P \) is the initial principal (initial value), \( r \) is the annual depreciation rate (as a decimal), and \( t \) is the time in years.
Here, \( P = 20700 \) dollars, \( r = 0.11 \) (since 11% = 0.11), and \( t = 11 \) years.

Step2: Substitute the values into the formula

Substitute \( P = 20700 \), \( r = 0.11 \), and \( t = 11 \) into the formula \( V = P(1 - r)^t \):

$$ V = 20700(1 - 0.11)^{11} $$

First, calculate \( 1 - 0.11 = 0.89 \). Then we have:

$$ V = 20700(0.89)^{11} $$

Step3: Calculate \( (0.89)^{11} \)

Using a calculator, \( (0.89)^{11} \approx 0.3138 \) (rounded to four decimal places).

Step4: Calculate the final value

Multiply \( 20700 \) by \( 0.3138 \):

$$ V \approx 20700 \times 0.3138 = 20700 \times 0.3138 = 6495.66 $$

Answer:

The value of the car after 11 years will be approximately \(\$6495.66\).