Question

Don buys a car valued at $25,000. when the car was new, it sold for $32,000. if the car depreciates exponentially at a rate of 7% per year, about how old is the car?

a) 2.6 years
b) 4.5 years
c) 4 years
d) 3 4 vears

Answer 1

To find the age of a car based on its depreciation, apply the exponential decay formula. Using the given current and original car values and the depreciation rate, solve for time. The car is approximately 3.4 years old, which rounds to 4 years, the closest answer. So the correct answer is option c.

Step-by-step explanation:

The student asks about determining the age of a car given its depreciation rate and its current value as compared to its original value. To solve this, we use the formula of exponential depreciation, which is:

V = P(1 - r)t,

where V is the current value of the car, P is the original price of the car when it was new, r is the rate of depreciation (expressed as a decimal), and t is the time in years.

In this case, we have:

V = $25,000

P = $32,000

r = 7% or 0.07

We need to solve for t.

First, divide both sides by P to get:

$25,000/$32,000 = (1 - 0.07)t

Then calculate:

0.78125 = 0.93t

Next, take the natural logarithm of both sides to solve for t:

ln(0.78125) = t * ln(0.93)

Finally, divide both sides by ln(0.93) to find t:

t = ln(0.78125)/ln(0.93) ≈ 3.4 years

Therefore, the closest answer would be (c) 4 years.