A new car is purchased for 16400 dollars. The value of the car depreciates at 6.75% per year. To the nearest year, how long will it be until the value of the car is 8500 dollars? Answer:

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A new car is purchased for 16400 dollars. The value of the car depreciates at  6.75% per year. To the nearest year, how long will it be until the value of the car is 8500 dollars? Answer:

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Identify Depreciation Type: Determine the type of depreciation. The car depreciates at a constant percentage each year. This is an example of exponential decay.Find Initial, Rate, Final Value: Identify the initial value (a), the rate of depreciation (r), and the final value (P). Initial value (a) = $16,400 Rate of depreciation (r) = 6.75% per year Final value (P) = $8,500Convert Rate to Decimal: Convert the rate of depreciation to a decimal. r = 6.75% = 0.0675Set Up Exponential Decay Formula: Set up the exponential decay formula. The formula for exponential decay is P = a(1 - r)^t, where P is the final amount, a is the initial amount, r is the rate of decay, and t is the time in years.Substitute Values into Formula: Substitute the known values into the formula. 8500 = 16400(1 - 0.0675)^tSolve for t: Solve for t. First, divide both sides by 16400 to isolate the exponential expression. 8500 / 16400 = (1 - 0.0675)^t 0.5183 ≈ (1 - 0.0675)^tTake Natural Logarithm: Take the natural logarithm of both sides to solve for t. ln(0.5183) ≈ ln((1 - 0.0675)^t)Use Logarithm Property: Use the property of logarithms that ln(a^b) = b*ln(a). ln(0.5183) ≈ t * ln(1 - 0.0675)Calculate Natural Logarithm: Calculate the natural logarithm of both sides. ln(0.5183) ≈ t * ln(0.9325)Divide by Logarithm: Divide both sides by ln(0.9325) to solve for t. t ≈ ln(0.5183) / ln(0.9325)Calculate Value of t: Use a calculator to find the value of t. t ≈ ln(0.5183) / ln(0.9325) t ≈ -0.6561 / -0.0693 t ≈ 9.47Round to Nearest Year: Round t to the nearest year. t ≈ 9 years

A new car is purchased for $16400$ dollars. The value of the car depreciates at $6.75 \%$ per year. To the nearest year, how long will it be until the value of the car is $8500$ dollars? $\newline$ Answer:

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A new car is purchased for $16400$ dollars. The value of the car depreciates at $6.75 \%$ per year. To the nearest year, how long will it be until the value of the car is $8500$ dollars? $\newline$ Answer: