javier has four cylindrical models. the heights, radii, and diagonals of the vertical cross-sections of the models are shown in the table.

model 1
radius: 14 cm
height: 48 cm
diagonal: 50 cm

model 2
radius: 6 cm
height: 35 cm
diagonal: 37 cm

model 3
radius: 20 cm
height: 40 cm
diagonal: 60 cm

model 4
radius: 24 cm
height: 9 cm
diagonal: 30 cm

Question

javier has four cylindrical models. the heights, radii, and diagonals of the vertical cross-sections of the models are shown in the table.

model 1
radius: 14 cm
height: 48 cm
diagonal: 50 cm

model 2
radius: 6 cm
height: 35 cm
diagonal: 37 cm

model 3
radius: 20 cm
height: 40 cm
diagonal: 60 cm

model 4
radius: 24 cm
height: 9 cm
diagonal: 30 cm

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Pythagorean Theorem" ], "new_concepts": [ "Cylinder Cross-Sections" ], "current_concepts": [ "Pythagorean Theorem", "Cylinder Cross-Sections" ] } ## Relate cylinder dimensions to the vertical cross-section diagonal $$ \begin{aligned} &\text{diameter } d = 2r \ &d^2 + h^2 = D^2 \end{aligned} $$ ## Verify the Pythagorean relation for each model $$ \begin{aligned} &\text{Model 1: } (2 \times 14)^2 + 48^2 = 28^2 + 48^2 = 784 + 2304 = 3088 eq 50^2 \ (2500) \ &\text{Model 2: } (2 \times 6)^2 + 35^2 = 12^2 + 35^2 = 144 + 1225 = 1369 = 37^2 \ (1369) \ &\text{Model 3: } (2 \times 20)^2 + 40^2 = 40^2 + 40^2 = 1600 + 1600 = 3200 eq 60^2 \ (3600) \ &\text{Model 4: } (2 \times 24)^2 + 9^2 = 48^2 + 9^2 = 2304 + 81 = 2385 eq 30^2 \ (900) \end{aligned} $$ ## Identify the correct model $$ \text{Model 2 satisfies the Pythagorean relationship: } 12^2 + 35^2 = 37^2 $$ Model 2 { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Geometry", "Pythagorean Theorem" ] }

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QUESTION IMAGE

Question

Explanation:

Relate cylinder dimensions to the vertical cross-section diagonal

Verify the Pythagorean relation for each model

Identify the correct model

Answer: