Question

George wanted to buy a car for $20,000. He knew he wanted to trade it in when the car had depreciated to $12,500. He knows that the car depreciates at a rate of 8.75%. Rounded to the nearest tenth of a year, how long would it be before he would have to trade in his car?

Answer 1

To determine how many years it would take for a $20,000 car to depreciate to $12,500 at a rate of 8.75% per year, an exponential decay formula is used, resulting in approximately 9.7 years.

Step-by-step explanation:

George wants to know how long it will take for a car worth $20,000 to depreciate to $12,500 at an annual depreciation rate of 8.75%. To solve this, we need to use the formula for exponential decay which is A = P(1 - r)^t, where A is the final amount, P is the initial principal balance, r is the rate of depreciation, and t is the time in years.

Let's substitute the values into the formula:

• P = $20,000 (initial value of the car)
• A = $12,500 (value of the car when George plans to trade it in)
• r = 0.0875 (depreciation rate in decimal form)

Now, we have $12,500 = $20,000(1 - 0.0875)^t. Simplify to get:

0.625 = (1 - 0.0875)^t

0.625 = 0.9125^t

We then take the logarithm of both sides:

• log(0.625) = t * log(0.9125)
• t = log(0.625) / log(0.9125)
• t ≈ 9.7 years

Therefore, it would take approximately 9.7 years for the car to depreciate to $12,500.