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{ "quality": "blurry", "question_count": 2, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Doubling Time", "Exponential Growth" ], "new_concepts": [], "current_concepts": [ "Doubling Time", "Exponential Growth" ] } ## State the exponential growth model Using the *Exponential Growth* and *Doubling Time* knowledge points $$ P(t) = P_0 \cdot 2^{t / T_d} $$ where: - $P_0 = 16,000$ is the initial population. - $T_d = 10$ years is the doubling time. - $t$ is the time in years. ## Calculate the population in 12 years Using the *Exponential Growth* and *Doubling Time* knowledge points $$ \begin{aligned} P(12) &= 16,000 \cdot 2^{12 / 10} \ &= 16,000 \cdot 2^{1.2} \ &\approx 16,000 \cdot 2.2973967 \ &\approx 36,758.35 \end{aligned} $$ Rounding to the nearest whole number gives $36,758$. ## Calculate the population in 24 years Using the *Exponential Growth* and *Doubling Time* knowledge points $$ \begin{aligned} P(24) &= 16,000 \cdot 2^{24 / 10} \ &= 16,000 \cdot 2^{2.4} \ &\approx 16,000 \cdot 5.2780316 \ &\approx 84,448.51 \end{aligned} $$ Rounding to the nearest whole number gives $84,449$. ## Question 1 What will the population be in 12 years? **36,758** ## Question 2 What will the population be in 24 years? **84,449** { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Algebra", "Exponential Growth" ] }

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State the exponential growth model

Calculate the population in 12 years

Calculate the population in 24 years

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Question 1

Question 2