Question

Sydney purchases a new car for $27,500. The value decreases at a rate of 15% each year. In approximately how many years will the car be worth $1,254?

A. 12 years
B. 14 years
C. 15 years
D. 16 years

Answer 1

It takes approximately 14 years for Sydney's car to depreciate to a value of $1,254 from an initial purchase price of $27,500 with an annual depreciation rate of 15%, by using the exponential decay formula.

Step-by-step explanation:

To calculate how long it will take for Sydney's car to depreciate to a value of $1,254 from an initial purchase price of $27,500 with an annual depreciation rate of 15%, we use the formula for exponential decay:

V = P (1 - r)^t

Where:

• V is the future value of the car.
• P is the present value of the car ($27,500).
• r is the rate of depreciation (15% or 0.15).
• t is the number of years.

Plugging the values we get:

1,254 = 27,500 (1 - 0.15)^t

Solving for t gives us:

1,254 = 27,500 (0.85)^t

Dividing both sides by 27,500 and taking the natural logarithm (ln) of both sides gives us:

ln(1,254 / 27,500) = t * ln(0.85)

Now compute the values:

ln(0.0456) = t * ln(0.85)

ln(0.0456) / ln(0.85) ≈ t

t ≈ ln(0.0456) / ln(0.85) ≈ 13.86

Since we are looking for an approximate number of years, we can round this up to say that it takes approximately 14 years for the car to be worth $1,254, making our answer option B.