devon purchased a new car valued at \$16,000 that depreciated continuously at a rate of 35%. its current value is \$2,000. the equation $2,000 = 16,000(1-r)^t$ represents the situation, where $t$ is the age of the car in years and $r$ is the rate of depreciation. about how old is devons car? use a calculator and round your answer to the nearest whole number.

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Question

devon purchased a new car valued at \$16,000 that depreciated continuously at a rate of 35%. its current value is \$2,000. the equation $2,000 = 16,000(1-r)^t$ represents the situation, where $t$ is the age of the car in years and $r$ is the rate of depreciation. about how old is devons car? use a calculator and round your answer to the nearest whole number.

1 year
2 years
5 years
8 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", "Solving Exponential Equations" ], "new_concepts": [], "current_concepts": [ "Solving Exponential Equations Algebraically", "Solving Exponential Equations", "Exponential Decay Modeling" ] } ## Substitute the given values into the equation Using the *Solving Exponential Equations* knowledge point $$ \begin{aligned} &r = 0.35\ &2,000 = 16,000(1 - 0.35)^t\ &2,000 = 16,000(0.65)^t \end{aligned} $$ ## Isolate the exponential term Using the *Solving Exponential Equations Algebraically* knowledge point $$ \begin{aligned} &\frac{2,000}{16,000} = (0.65)^t\ &0.125 = (0.65)^t \end{aligned} $$ ## Solve for t using logarithms Using the *Solving Exponential Equations Algebraically* knowledge point $$ \begin{aligned} &\ln(0.125) = \ln((0.65)^t)\ &\ln(0.125) = t \cdot \ln(0.65)\ &t = \frac{\ln(0.125)}{\ln(0.65)} \end{aligned} $$ ## Calculate the numerical value of t Using the *Solving Exponential Equations* knowledge point $$ \begin{aligned} &t \approx \frac{-2.07944}{-0.43078}\ &t \approx 4.827\ &t \approx 5 \end{aligned} $$

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

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QUESTION IMAGE

Question

Explanation:

Substitute the given values into the equation

Isolate the exponential term

Solve for t using logarithms

Calculate the numerical value of t

Answer: