A new car is purchased for $42,000 and over time its value depreciates by one half every 3 years. What is the value of the car 10 years after it was purchased, to the nearest hundred dollars? Answer:

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A new car is purchased for  $42,000 and over time its value depreciates by one half every 3 years. What is the value of the car 10 years after it was purchased, to the nearest hundred dollars? Answer:

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Initial Value and Depreciation Rate: Determine the initial value of the car and the rate of depreciation. The initial value of the car is $42,000, and it depreciates by half every 3 years. This is an exponential decay problem.Find a and b: Find the values of a and b. a = $42,000 (initial value of the car) b = 1/2 (the car loses half its value every 3 years)Calculate Number of Halvings: Calculate x, the number of times the car's value is halved in 10 years. t = 10 (number of years after purchase) T = 3 (number of years it takes for the car's value to halve) x = t/T x = 10/3 Since the value halves every 3 years, we need to round x to the nearest whole number because the car can't partially halve its value. x ≈ 3 (after 9 years, not 10)Exponential Decay Formula: Use the exponential decay formula to calculate the car's value after 10 years. Exponential Decay: V(x) = a(b)^(x) Which equation represents the exponential decay? V(x) = a(b)^(x) Substitute $42,000 for a, 1/2 for b, and 3 for x V(x) = 42000(1/2)^(3)Evaluate Car's Value at 9 Years: Evaluate the expression to find the car's value after 9 years (since we rounded down to 3 halvings). V(x) = 42000(1/2)^(3) V(x) = 42000 * 1/8 V(x) = 5250 The value of the car after 9 years is $5,250.Estimate Car's Value at 10 Years: Since we calculated the value at 9 years instead of 10, we need to estimate the value at 10 years. We know that the car's value is not halving between the 9th and 10th year, so the value at 10 years will be slightly less than $5,250. To estimate, we can assume a linear depreciation over the next year, which would be less than half of the yearly depreciation rate from the previous intervals. However, since we need to round to the nearest hundred dollars, and the value at 9 years is already a multiple of $100, we can conclude that the value at 10 years, to the nearest hundred dollars, will still be $5,200.

A new car is purchased for $\$ 42,000$ and over time its value depreciates by one half every $3$ years. What is the value of the car $10$ years after it was purchased, to the nearest hundred dollars? $\newline$ Answer:

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A new car is purchased for $\$ 42,000$ and over time its value depreciates by one half every $3$ years. What is the value of the car $10$ years after it was purchased, to the nearest hundred dollars? $\newline$ Answer: