a type of plant is introduced into an ecosystem and quickly begins to take over. a scientist counts the number of plants after (m) months and develops the equation (p(m) = 19.3(1.089)^m) to model the situation. most recently, the scientist counted 138 plants. assuming there are no limiting factors to the growth of the plants, about how many months have passed since the plants were first introduced?

6.1
6.6
7.2
23.1

Question

a type of plant is introduced into an ecosystem and quickly begins to take over. a scientist counts the number of plants after (m) months and develops the equation (p(m) = 19.3(1.089)^m) to model the situation. most recently, the scientist counted 138 plants. assuming there are no limiting factors to the growth of the plants, about how many months have passed since the plants were first introduced?

6.1
6.6
7.2
23.1

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", "Logarithmic Evaluation" ] } ## Set up the exponential equation Using the *Solving Exponential Equations* knowledge point $$ \begin{aligned} P(m) &= 19.3(1.089)^m \ 138 &= 19.3(1.089)^m \end{aligned} $$ ## Isolate the exponential term Using the *Solving Exponential Equations Algebraically* knowledge point $$ \begin{aligned} (1.089)^m &= \frac{138}{19.3} \ (1.089)^m &\approx 7.15026 \end{aligned} $$ ## Solve for m using logarithms Using the *Solving Exponential Equations Algebraically* knowledge point $$ \begin{aligned} \ln(1.089)^m &= \ln(7.15026) \ m \ln(1.089) &= \ln(7.15026) \ m &= \frac{\ln(7.15026)}{\ln(1.089)} \ m &\approx \frac{1.96715}{0.08526} \approx 23.07 \end{aligned} $$ ## Round to the nearest tenth Using the *Solving Exponential Equations Algebraically* knowledge point $$ m \approx 23.1 $$

(A) 6.1

(B) 6.6

(C) 7.2

(D) 23.1

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

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QUESTION IMAGE

Question

Explanation:

Set up the exponential equation

Isolate the exponential term

Solve for m using logarithms

Round to the nearest tenth

Answer: