two different cars each depreciate to 60% of their respective original values. the first car depreciates at an annual rate of 10%. the second car depreciates at an annual rate of 15%. what is the approximate difference in the ages of the two cars?

1.7 years
2.0 years
3.1 years
5.0 years

Question

two different cars each depreciate to 60% of their respective original values. the first car depreciates at an annual rate of 10%. the second car depreciates at an annual rate of 15%. what is the approximate difference in the ages of the two cars?

1.7 years
2.0 years
3.1 years
5.0 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 Decay", "Logarithmic Equations" ], "new_concepts": [], "current_concepts": [ "Exponential Decay", "Logarithmic Equations" ] } ## Set up exponential decay equations for both cars $$ \begin{aligned} (1 - 0.10)^{t_1} &= 0.60 \implies 0.9^{t_1} = 0.6 \ (1 - 0.15)^{t_2} &= 0.60 \implies 0.85^{t_2} = 0.6 \end{aligned} $$ ## Solve for the ages using logarithms $$ \begin{aligned} t_1 &= \frac{\ln(0.6)}{\ln(0.9)} \approx \frac{-0.5108}{-0.1054} \approx 4.847 \text{ years} \ t_2 &= \frac{\ln(0.6)}{\ln(0.85)} \approx \frac{-0.5108}{-0.1625} \approx 3.143 \text{ years} \end{aligned} $$ ## Calculate the difference in ages $$ \Delta t = t_1 - t_2 \approx 4.847 - 3.143 = 1.704 \text{ years} $$

(A) 1.7 years

(B) 2.0 years

(C) 3.1 years

(D) 5.0 years

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

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QUESTION IMAGE

Question

Explanation:

Set up exponential decay equations for both cars

Solve for the ages using logarithms

Calculate the difference in ages

Answer: