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A new car is purchased for 20700 dollars. The value of the car depreciates at 7.25% per year. To the nearest year, how long will it be until the value of the car is 10800 dollars

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Final answer:

To find the time until the car depreciates to $10,800, we use the exponential decay formula. After solving the equation, we find that it will take approximately 8 years for the car to depreciate to that value.

Step-by-step explanation:

The question asks how long it will take for a car, purchased for $20,700, to depreciate to $10,800 at an annual depreciation rate of 7.25%. This is a typical exponential decay problem, where we can use the formula for exponential decay to find the time, t:

V = P(1 - r)^t

Where:

  • V is the future value of the car
  • P is the initial value of the car
  • r is the depreciation rate (as a decimal)
  • t is the time in years

Plugging in the given values:

10800 = 20700(1 - 0.0725)^t

To find t, we need to solve for it using logarithms:

  1. Divide both sides by 20700:
    (10800 / 20700) = (1 - 0.0725)^t
  2. Calculate the left side and take the natural logarithm of both sides:
    ln(10800 / 20700) = ln((1 - 0.0725)^t)
  3. Apply the power rule of logarithms to move t out front:
    t * ln(1 - 0.0725) = ln(10800 / 20700)
  4. Solve for t:
    t = ln(10800 / 20700) / ln(1 - 0.0725)
  5. Calculate the value of t using a calculator.

When you do the math, you will find that t is approximately 8 years (rounding to the nearest year).

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