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A new car is purchased for 15300 dollars. The value of the car depreciates at
14.25% per year. What will the value of the car be, to the nearest cent, after 6 years?

A new car is purchased for $15300$ dollars. The value of the car depreciates at $14.25 \%$ per year. What will the value of the car be, to the nearest cent, after $6$ years?

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Exponential growth and decay: word problems

Full solution

Q. A new car is purchased for

15300

dollars. The value of the car depreciates at

14.25 \%

per year. What will the value of the car be, to the nearest cent, after

6

years?

Calculate Depreciation: To calculate the depreciation, we can use the formula for exponential decay: $V = P(1 - r)^t$ , where $V$ is the final value, $P$ is the initial value, $r$ is the rate of depreciation, and $t$ is the time in years.
Identify Values: Let's identify the values of $P$ , $r$ , and $t$ . The initial value of the car $P$ is $\$15,300$ , the rate of depreciation $r$ is $14.25\%$ or $0.1425$ when converted to a decimal, and the time $t$ is $6$ years.
Substitute Values: Now we can substitute these values into the formula to calculate the car's value after $6$ years: $V = 15300(1 - 0.1425)^6$ .
Calculate Inside Parentheses: Calculate the value inside the parentheses first: $1 - 0.1425 = 0.8575$ .
Raise to Power: Now raise $0.8575$ to the power of $6$ : $(0.8575)^6 \approx 0.4088$ (rounded to four decimal places for precision in intermediate steps).
Multiply Initial Value: Multiply the initial value of the car by the result from the previous step: $V = 15300 \times 0.4088 \approx 6254.64$ .
Round Final Value: Round the final value to the nearest cent: The value of the car after $6$ years is approximately $\$6,254.64$ .

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