Final answer:
To determine how many years Charles should use his car before selling it, we can use the formula for exponential decay. By plugging in the given values and solving for t, we find that Charles should use this car for approximately 13 years.
Step-by-step explanation:
To determine how many years Charles should use his car before selling it, we can set up an equation using the concept of exponential decay.
The value of Charles' car after a certain number of years can be calculated using the formula: V = P imes (1 - r)^t, where V is the value of the car after t years, P is the initial purchase price, r is the rate of decline (expressed as a decimal), and t is the number of years.
In this case, we know that Charles wants to sell the car for at least $6,000, so the value of the car after t years should be equal to or greater than $6,000. Plugging in the given values, we have: 6,000 = 9,520 imes (1 - 0.05)^t.
To solve for t, we need to isolate it. Dividing both sides of the equation by 9,520 gives: (1 - 0.05)^t = 6,000 / 9,520. Taking the natural logarithm (ln) of both sides: t imes ln(0.95) = ln(6,000/9,520).
Using a calculator, we can solve for t. Approximating ln(0.95) = -0.051293, we have: t imes -0.051293 = -0.650523. Dividing both sides by -0.051293: t = -0.650523 / -0.051293 ≈ 12.68.
Rounding up to the nearest whole number, we get t = 13.