Question

an equation for the depreciation of a car is given by (y = a(1 - r)^t), where (y) = current value of the car, (a) = original cost, (r) = rate of depreciation, and (t) = time, in years. the current value of a car is $12,282.50. the car originally cost $20,000 and depreciates at a rate of 15% per year. how old is the car?

(\frac{1}{4}) years

(\frac{1}{3}) years

3 years

4 years

Explanation:

Identify given parameters

Using the Exponential Decay Modeling knowledge point

$$
LATEXBLOCK0
$$

Set up the equation

Using the Exponential Decay Modeling knowledge point

$$ 12282.50 = 20000(1 - 0.15)^t $$

Simplify the base and isolate the exponential term

Using the Exponential Decay Modeling knowledge point

$$
LATEXBLOCK1
$$

Solve for t using logarithms

Using the Solving Exponential Equations Algebraically knowledge point

$$
LATEXBLOCK2
$$

Answer:

• (A) \(\frac{1}{4}\) years
• (B) \(\frac{1}{3}\) years
• (C) 3 years (Correct answer)
• (D) 4 years