Question

devon purchased a new car valued at \\$16,000 that depreciated continuously at a rate of 35\\%. its current value is \\$2,000. the equation \\(2,000 = 16,000(1-r)^t\\) represents the situation, where \\(t\\) is the age of the car in years and \\(r\\) is the rate of depreciation. about how old is devons car? use a calculator and round your answer to the nearest whole number.

1 year
2 years
5 years
8 years

Explanation:

Identify given values

We are given the equation:

$$2,000 = 16,000(1-r)^t$$

The rate of depreciation is:

$$r = 35\% = 0.35$$

Substitute the rate

Substitute \(r = 0.35\) into the equation:

$$2,000 = 16,000(1-0.35)^t$$

$$2,000 = 16,000(0.65)^t$$

Isolate the exponential term

Using the Exponential Equations knowledge point, divide both sides by \(16,000\):

$$\frac{2,000}{16,000} = (0.65)^t$$

$$0.125 = (0.65)^t$$

Solve for t using logarithms

Take the natural logarithm of both sides:

$$\ln(0.125) = \ln(0.65^t)$$

$$\ln(0.125) = t \cdot \ln(0.65)$$

$$t = \frac{\ln(0.125)}{\ln(0.65)}$$

Calculate and round the value

Compute the numerical value:

$$t \approx \frac{-2.0794}{-0.4308} \approx 4.83$$

Rounding to the nearest whole number gives:

$$t \approx 5\text{ years}$$

Answer:

• 1 year
• 2 years
• 5 years (Correct answer)
• 8 years