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

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

(\frac{1}{4}) years

(\frac{1}{3}) years

3 years

4 years

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Exponential Modeling", "Exponential Equations" ], "new_concepts": [], "current_concepts": [ "Exponential Modeling", "Exponential Equations", "Logarithmic Equations" ] } ## Substitute the given values into the depreciation equation $$ \begin{aligned} y &= 12282.50 \ A &= 20000 \ r &= 0.15 \ 12282.50 &= 20000(1 - 0.15)^t \ 12282.50 &= 20000(0.85)^t \end{aligned} $$ ## Isolate the exponential term $$ \begin{aligned} (0.85)^t &= \frac{12282.50}{20000} \ (0.85)^t &= 0.614125 \end{aligned} $$ ## Solve for t using logarithms $$ \begin{aligned} t &= \frac{\ln(0.614125)}{\ln(0.85)} \ t &\approx \frac{-0.48752}{-0.162518} \ t &\approx 3 \end{aligned} $$

(A) $\frac{1}{4}$ years

(B) $\frac{1}{3}$ years

(C) 3 years

(D) 4 years

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Exponential Equations" ] }

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QUESTION IMAGE

Question

Explanation:

Substitute the given values into the depreciation equation

Isolate the exponential term

Solve for t using logarithms

Answer: