{"id":"cb_question_bank:224e737b-b611-442b-af26-ef476da15a5b","source_id":"cb_question_bank","source_item_id":"224e737b-b611-442b-af26-ef476da15a5b","external_id":"c1c7e4d7-d6dc-4e54-8b49-cbd88eab7153","ibn":null,"question_id":"a07ed090","program":"SAT","module":"math","domain_code":"S","domain":"Geometry and Trigonometry","skill_code":"S.A.","skill":"Area and volume","difficulty":"H","score_band":6,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"The figure shown is a right circular cylinder with a radius of r and height of h . A second right circular cylinder (not shown) has a volume that is 392 times as large as the volume of the cylinder shown. Which of the following could represent the radius upper R , in terms of r , and the height upper H , in terms of h , of the second cylinder?","stimulus":"","stimulus_html":null,"stem_html":"

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The figure shown is a right circular cylinder with a radius of r\n and height of h\n. A second right circular cylinder (not shown) has a volume that is 392\n times as large as the volume of the cylinder shown. Which of the following could represent the radius R\n, in terms of r\n, and the height H\n, in terms of h\n, of the second cylinder?

","rationale_html":"

Choice C is correct. The volume of a right circular cylinder is equal to πa2b, where a\n is the radius of a base of the cylinder and b\n is the height of the cylinder. It’s given that the cylinder shown has a radius of r\n and a height of h\n. It follows that the volume of the cylinder shown is equal to πr2h. It’s given that the second right circular cylinder has a radius of R\n and a height of H\n. It follows that the volume of the second cylinder is equal to πR2H. Choice C gives \n\tR\n\t=\n\t\n\t\t7\n\t\tr\n\t\n\n and \n\tH\n\t=\n\t\n\t\t8\n\t\th\n\t\n\n. Substituting \n\t7\n\tr\n\n for R\n and \n\t8\n\th\n\n for H\n in the expression that represents the volume of the second cylinder yields π7r28h, or π49r28h, which is equivalent to π392r2h, or 392πr2h. This expression is equal to 392\n times the volume of the cylinder shown, πr2h. Therefore, \n\tR\n\t=\n\t\n\t\t7\n\t\tr\n\t\n\n and \n\tH\n\t=\n\t\n\t\t8\n\t\th\n\t\n\n could represent the radius R\n, in terms of r\n, and the height H\n, in terms of h\n, of the second cylinder.

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Choice A is incorrect. Substituting \n\t8\n\tr\n\n for R\n and \n\t7\n\th\n\n for H\n in the expression that represents the volume of the second cylinder yields π8r27h, or π64r27h, which is equivalent to π448r2h, or 448πr2h. This expression is equal to 448\n, not 392\n, times the volume of the cylinder shown. 

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Choice B is incorrect. Substituting \n\t8\n\tr\n\n for R\n and \n\t49\n\th\n\n for H\n in the expression that represents the volume of the second cylinder yields π8r249h, or π64r249h, which is equivalent to π3,136r2h, or 3,136πr2h. This expression is equal to 3,136\n, not 392\n, times the volume of the cylinder shown.

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Choice D is incorrect. Substituting \n\t49\n\tr\n\n for R\n and \n\t8\n\th\n\n for H\n in the expression that represents the volume of the second cylinder yields π49r28h, or π2,401r28h, which is equivalent to π19,208r2h, or 19,208πr2h. This expression is equal to 19,208\n, not 392\n, times the volume of the cylinder shown.

","correct_answer":["C"],"has_media":true,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.837000+00:00","source_updated_at":"2023-08-02T20:25:59.837000+00:00","options":[{"label":"A","content_html":"

\n\tR\n\t=\n\t\n\t\t8\n\t\tr\n\t\n\n and \n\tH\n\t=\n\t\n\t\t7\n\t\th\n\t\n\n

","ord":0},{"label":"B","content_html":"

\n\tR\n\t=\n\t\n\t\t8\n\t\tr\n\t\n\n and \n\tH\n\t=\n\t\n\t\t49\n\t\th\n\t\n\n

","ord":1},{"label":"C","content_html":"

\n\tR\n\t=\n\t\n\t\t7\n\t\tr\n\t\n\n and \n\tH\n\t=\n\t\n\t\t8\n\t\th\n\t\n\n

","ord":2},{"label":"D","content_html":"

\n\tR\n\t=\n\t\n\t\t49\n\t\tr\n\t\n\n and \n\tH\n\t=\n\t\n\t\t8\n\t\th\n\t\n\n

","ord":3}],"attribution":{"source_id":"cb_question_bank","source_name":"College Board digital SAT question bank (community dump)","rights_holder":"College Board","canonical_url":"https://satsuitequestionbank.collegeboard.org/","retrieved_from":"https://raw.githubusercontent.com/mdn522/sat-question-bank/main/data/cb-digital-questions.json","attribution":"SAT practice questions © College Board, from the official SAT Suite Question Bank. Retrieved via the community dump at github.com/mdn522/sat-question-bank.","disclaimer":"SAT® and PSAT/NMSQT® are trademarks registered by the College Board, which is not affiliated with, does not sponsor, and does not endorse this project.","license":"College Board copyright; source repository carries no licence. Local use only.","redistributable":false,"item_count":2017}}