Kelly bought a used car for $20,000, and the dealer informed her it would depreciate by 9% annually after the purchase. Assuming the value continues to decline at that rate, calculate the time it will take for the car's worth to reduce to $7,000. Round your answer to the nearest tenth of a year.

To calculate the time it will take for the car's value to reduce to $7,000 from $20,000, with an annual depreciation of 9%, we can use the formula for exponential decay, which is:

V = P * (1 - r)^t

Where: V is the future value of the car (in this case, $7,000), P is the present value or initial amount (in this case, $20,000), r is the rate of depreciation (in this case, 9% or 0.09 as a decimal), t is the time in years.

We want to find t, so we rearrange the formula to solve for t:

t = log(V/P) / log(1 - r)

Plugging in the numbers:

t = log(7000/20000) / log(1 - 0.09) t = log(0.35) / log(0.91) t ≈ log(0.35) / log(0.91) t ≈ -0.455931955 / -0.04575749056

Now, if we divide the two values we get:

t ≈ -0.455931955 / -0.04575749056 ≈ 9.971

Rounded to the nearest tenth of a year, the time it will take for the car's value to reduce to $7,000 is approximately:

t ≈ 10.0 years