the radius of the large sphere is double the radius of the small sphere.
how many times is the volume of the large sphere than the small sphere?
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Question

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": [ "Volume of a Sphere" ], "new_concepts": [ "Volume Scaling of Solids", "Similar Solids Ratio" ], "current_concepts": [ "Volume of a Sphere", "Volume Scaling of Solids", "Similar Solids Ratio" ] } ## Define the given radii Using the *Volume of a Sphere* knowledge point $$ \begin{aligned} &r_{\text{small}} = 3\ &r_{\text{large}} = 6 \end{aligned} $$ ## Calculate the volume of each sphere Using the *Volume of a Sphere* knowledge point $$ \begin{aligned} &V_{\text{small}} = \frac{4}{3}\pi (3)^3 = \frac{4}{3}\pi (27) = 36\pi\ &V_{\text{large}} = \frac{4}{3}\pi (6)^3 = \frac{4}{3}\pi (216) = 288\pi \end{aligned} $$ ## Find the ratio of the volumes Using the *Volume of a Sphere* knowledge point $$ \begin{aligned} &\text{Ratio} = \frac{V_{\text{large}}}{V_{\text{small}}} = \frac{288\pi}{36\pi} = 8 \end{aligned} $$

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{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Volume of a Sphere" ] }

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Question

Explanation:

Define the given radii

Calculate the volume of each sphere

Find the ratio of the volumes

Answer: