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{ "quality": "blurry", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Solving Exponential Equations Algebraically", "Solving Exponential Equations Graphically" ], "new_concepts": [], "current_concepts": [ "Solving Exponential Equations Algebraically", "Solving Exponential Equations Graphically", "Exponential Decay Modeling" ] } ## Isolate the exponential term Using the *Solving Exponential Equations Algebraically* knowledge point $$ \begin{aligned} 364 &= 900(1 - 0.14)^t \ 364 &= 900(0.86)^t \ (0.86)^t &= \frac{364}{900} \ (0.86)^t &\approx 0.4044 \end{aligned} $$ ## Solve for t using logarithms Using the *Solving Exponential Equations* knowledge point $$ \begin{aligned} t &= \log_{0.86}(0.4044) \ t &= \frac{\ln(0.4044)}{\ln(0.86)} \ t &\approx \frac{-0.9053}{-0.1508} \ t &\approx 6.003 \end{aligned} $$ ## Round to the nearest whole year Using the *Exponential Decay Modeling* knowledge point $$ t \approx 6 $$ A computer was originally purchased for $900 and undergoes a 14% annual depreciation rate. The computer is now worth $364. The number of years that have passed, $t$, since the computer was purchased can be found using the following equation: $$364 = 900(1 - 0.14)^t$$ Solve the equation by graphing to determine how many years have passed since the computer was purchased. Round to the nearest whole year. $t \approx$ 6 years { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Solving Exponential Equations" ] }

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Explanation:

Isolate the exponential term

Solve for t using logarithms

Round to the nearest whole year

Answer: