select the correct answer from each drop-down menu.

if a cone has the same radius and height as a cylinder, the volume of the cone is choose an answer the volume of the cylinder. if a cylinder and a sphere have the same radius and the cylinders height is twice its radius, then the volume of the sphere is choose an answer the volume of the cylinder.

Question

select the correct answer from each drop-down menu.

if a cone has the same radius and height as a cylinder, the volume of the cone is choose an answer the volume of the cylinder. if a cylinder and a sphere have the same radius and the cylinders height is twice its radius, then the volume of the sphere is choose an answer the volume of the cylinder.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Volume of Cylinder", "Volume of Cone", "Geometric Volume Relations" ], "new_concepts": [ "Volume of Sphere" ], "current_concepts": [ "Volume of Cylinder", "Volume of Cone", "Volume of Sphere", "Geometric Volume Relations" ] } ## Compare cone and cylinder volumes $$ \begin{aligned} V_{\text{cylinder}} &= \pi r^2 h \ V_{\text{cone}} &= \frac{1}{3}\pi r^2 h = \frac{1}{3} V_{\text{cylinder}} \end{aligned} $$ ## Compare sphere and cylinder volumes $$ \begin{aligned} h &= 2r \ V_{\text{cylinder}} &= \pi r^2 (2r) = 2\pi r^3 \ V_{\text{sphere}} &= \frac{4}{3}\pi r^3 = \frac{2}{3} (2\pi r^3) = \frac{2}{3} V_{\text{cylinder}} \end{aligned} $$ If a cone has the same radius and height as a cylinder, the volume of the cone is one-third the volume of the cylinder. If a cylinder and a sphere have the same radius and the cylinder's height is twice its radius, then the volume of the sphere is two-thirds the volume of the cylinder. { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Geometry", "Geometric Volume Relations" ] }

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Question

Explanation:

Compare cone and cylinder volumes

Compare sphere and cylinder volumes

Answer: