a non-food crop is infected by pests on the $1^{\text{st}}$ of a month. the pests infect the crop in such a way that the area infected doubles after each month. if the pests continue to infect the crop in this way, the non-food crop will be entirely infected after the sixth month.

after which month will one-eighth of the non-food crop be infected?

- one-eighth of the non-food crop will be infected after the second month.
- one-eighth of the non-food crop will be infected after the fourth month.
- one-eighth of the non-food crop will be infected after the fifth month.
- one-eighth of the non-food crop will be infected after the third month.

Question

a non-food crop is infected by pests on the $1^{\text{st}}$ of a month. the pests infect the crop in such a way that the area infected doubles after each month. if the pests continue to infect the crop in this way, the non-food crop will be entirely infected after the sixth month.

after which month will one-eighth of the non-food crop be infected?

- one-eighth of the non-food crop will be infected after the second month.
- one-eighth of the non-food crop will be infected after the fourth month.
- one-eighth of the non-food crop will be infected after the fifth month.
- one-eighth of the non-food crop will be infected after the third month.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Exponential Functions" ], "new_concepts": [ "Exponential Decay", "Geometric Growth" ], "current_concepts": [ "Exponential Functions", "Exponential Decay", "Geometric Growth" ] } ## Define the exponential growth model $$ \begin{aligned} &A(t) = A_0 \cdot 2^t \ &A(6) = 1 \ &A_0 \cdot 2^6 = 1 \implies A_0 = \frac{1}{64} \end{aligned} $$ ## Solve for the target fraction $$ \begin{aligned} &A(t) = \frac{1}{8} \ &\frac{1}{64} \cdot 2^t = \frac{1}{8} \ &2^t = 8 \implies 2^t = 2^3 \implies t = 3 \end{aligned} $$ ## Determine the corresponding month $$ \begin{aligned} &\text{Since } t = 3 \text{ represents the number of months elapsed,} \ &\text{one-eighth of the crop is infected after the third month.} \end{aligned} $$

One-eighth of the non-food crop will be infected after the second month.

One-eighth of the non-food crop will be infected after the third month.

One-eighth of the non-food crop will be infected after the fourth month.

One-eighth of the non-food crop will be infected after the fifth month.

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

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QUESTION IMAGE

Question

Explanation:

Define the exponential growth model

Solve for the target fraction

Determine the corresponding month

Answer: