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", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Exponential Decay Modeling", "Solving Exponential Equations Algebraically" ], "new_concepts": [], "current_concepts": [ "Exponential Decay Modeling", "Solving Exponential Equations Algebraically" ] } ## Model the depreciation of the first car Using the *Exponential Decay Modeling* knowledge point $$ \begin{aligned} 0.60 &= (1 - 0.10)^{t_1} \ 0.60 &= 0.90^{t_1} \end{aligned} $$ ## Solve for the age of the first car Using the *Solving Exponential Equations Algebraically* knowledge point $$ \begin{aligned} t_1 &= \frac{\ln(0.60)}{\ln(0.90)} \ t_1 &\approx \frac{-0.5108}{-0.1054} \approx 4.846\text{ years} \end{aligned} $$ ## Model the depreciation of the second car Using the *Exponential Decay Modeling* knowledge point $$ \begin{aligned} 0.60 &= (1 - 0.15)^{t_2} \ 0.60 &= 0.85^{t_2} \end{aligned} $$ ## Solve for the age of the second car Using the *Solving Exponential Equations Algebraically* knowledge point $$ \begin{aligned} t_2 &= \frac{\ln(0.60)}{\ln(0.85)} \ t_2 &\approx \frac{-0.5108}{-0.1625} \approx 3.143\text{ years} \end{aligned} $$ ## Calculate the difference in ages $$ \begin{aligned} \Delta t &= t_1 - t_2 \ \Delta t &\approx 4.846 - 3.143 = 1.703\text{ years} \end{aligned} $$

(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 Modeling" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Model the depreciation of the first car

Solve for the age of the first car

Model the depreciation of the second car

Solve for the age of the second car

Calculate the difference in ages

Answer: