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 value of a car is half what it originally cost. the rate of depreciation is 10%. approximately how old is the car?

3.3 years
5.0 years
5.6 years
6.6 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 value of a car is half what it originally cost. the rate of depreciation is 10%. approximately how old is the car?

3.3 years
5.0 years
5.6 years
6.6 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": [ "Solving Exponential Equations Algebraically", "Exponential Decay Modeling" ], "new_concepts": [], "current_concepts": [ "Exponential Decay Modeling", "Solving Exponential Equations Algebraically" ] } ## Set up the exponential decay equation Using the *Exponential Decay Modeling* knowledge point $$ \begin{aligned} &y = A(1 - r)^t \ &y = 0.5A \ &r = 0.10 \ &0.5A = A(1 - 0.10)^t \ &0.5 = 0.9^t \end{aligned} $$ ## Solve for time t algebraically Using the *Solving Exponential Equations Algebraically* knowledge point $$ \begin{aligned} &\ln(0.5) = \ln(0.9^t) \ &\ln(0.5) = t \ln(0.9) \ &t = \frac{\ln(0.5)}{\ln(0.9)} \ &t \approx \frac{-0.69315}{-0.10536} \approx 6.5788 \end{aligned} $$ ## Match with the closest option The calculated value of $t \approx 6.6$ years matches the fourth option.

(A) 3.3 years

(B) 5.0 years

(C) 5.6 years

(D) 6.6 years

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

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QUESTION IMAGE

Question

Explanation:

Set up the exponential decay equation

Solve for time t algebraically

Match with the closest option

Answer: