the cylinder below has a volume of (785\text{ ft}^3), and the cone has a volume of (314\text{ ft}^3).

which explains which figure has the greater height?

- the cylinder has a greater height because the bases are the same, but the cylinder has a greater volume.
- the cylinder has a greater height because the volume of the cylinder is greater than twice the volume of the cone.
- the cone has a greater height because the bases are the same, but the volume of the cylinder is less than 3 times the volume of the cone.
- the cone has a greater height because the bases are the same, but the volume of the cylinder is less than 4 times the volume of the cone.

Question

the cylinder below has a volume of (785\text{ ft}^3), and the cone has a volume of (314\text{ ft}^3).

which explains which figure has the greater height?

- the cylinder has a greater height because the bases are the same, but the cylinder has a greater volume.
- the cylinder has a greater height because the volume of the cylinder is greater than twice the volume of the cone.
- the cone has a greater height because the bases are the same, but the volume of the cylinder is less than 3 times the volume of the cone.
- the cone has a greater height because the bases are the same, but the volume of the cylinder is less than 4 times the volume of the cone.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Cylinder Geometry", "Volume of a Cone", "Volume Relationship between Cylinder and Cone" ], "new_concepts": [], "current_concepts": [ "Cylinder Geometry", "Volume of a Cone", "Volume Relationship between Cylinder and Cone" ] } ## Calculate the height of the cylinder $$ \begin{aligned} V_{\text{cylinder}} &= B \cdot h_{\text{cylinder}} \ 785 &= 78.5 \cdot h_{\text{cylinder}} \ h_{\text{cylinder}} &= \frac{785}{78.5} = 10\text{ ft} \end{aligned} $$ ## Calculate the height of the cone $$ \begin{aligned} V_{\text{cone}} &= \frac{1}{3} B \cdot h_{\text{cone}} \ 314 &= \frac{1}{3} (78.5) \cdot h_{\text{cone}} \ h_{\text{cone}} &= \frac{314 \cdot 3}{78.5} = \frac{942}{78.5} = 12\text{ ft} \end{aligned} $$ ## Compare heights and evaluate options $$ \begin{aligned} &h_{\text{cone}} = 12 > h_{\text{cylinder}} = 10 \ &V_{\text{cylinder}} = 785,\quad 3 \cdot V_{\text{cone}} = 3 \cdot 314 = 942 \ &785 < 942 \implies V_{\text{cylinder}} < 3 \cdot V_{\text{cone}} \end{aligned} $$

The cylinder has a greater height because the bases are the same, but the cylinder has a greater volume.

The cylinder has a greater height because the volume of the cylinder is greater than twice the volume of the cone.

The cone has a greater height because the bases are the same, but the volume of the cylinder is less than 3 times the volume of the cone.

The cone has a greater height because the bases are the same, but the volume of the cylinder is less than 4 times the volume of the cone.

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Volume Relationship between Cylinder and Cone" ] }

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QUESTION IMAGE

Question

Explanation:

Calculate the height of the cylinder

Calculate the height of the cone

Compare heights and evaluate options

Answer: