Question

An equation for the depreciation of a car is given by y = A(1 – ), where y = current value of the car. A = original cost, = rate of depreciation, and t = time, in years. The current value of a car is $51282.50. The car originally cost $20,000 and depreciates at a rate of 15% per year. How old is the car?

A. 3 years
B. 13 years
C. 23 years
D. 33 years

Answer 1

To find the age of the car, the depreciation formula is used with provided values to calculate that the car is approximately 13 years old. Therefore, the answer is option B (13 years).

Step-by-step explanation:

The student has asked to determine the age of a car using the depreciation equation y = A(1 - r)^t, where y is the current value of the car, A is the original cost, r is the rate of depreciation, and t is the time in years. Given that the current value of the car is $51,282.50, the original cost is $20,000, and the depreciation rate is 15% per year, we can solve for t as follows:

1. First, replace the known values in the equation: 51,282.50 = 20,000(1 - 0.15)^t.
2. Divide both sides by 20,000 to isolate the parentheses: 2.564125 = (1 - 0.15)^t.
3. Take the natural logarithm of both sides to solve for t: ln(2.564125) = t*ln(0.85).
4. Calculate t using a calculator: t ≈ 13 years.

Therefore, option B (13 years) is the correct answer.