{"id":"cb_question_bank:639e9d39-694d-4f39-9654-b53995d7c795","source_id":"cb_question_bank","source_item_id":"639e9d39-694d-4f39-9654-b53995d7c795","external_id":null,"ibn":"06496-DC","question_id":"b0dc920d","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":"A manufacturer determined that right cylindrical containers with a height that is 4 inches longer than the radius offer the optimal number of containers to be displayed on a shelf. Which of the following expresses the volume, V , in cubic inches, of such containers, where r is the radius, in inches?","stimulus":"","stimulus_html":null,"stem_html":"<p class=\"stem_paragraph \">A manufacturer determined that right cylindrical containers with a height that is 4 inches longer than the radius offer the optimal number of containers to be displayed on a shelf. Which of the following expresses the volume, <span class=\"italic\">V</span>, in cubic inches, of such containers, where <span class=\"italic\">r</span> is the radius, in inches?</p>\n","rationale_html":"<p>Choice D is correct. The volume, <span class=\"italic\">V</span>, of a right cylinder is given by the formula <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/5a750e8de0d2c8dfee439ab17b14523361fdb26d14f609112027eee3efb10ed3\" alt=\"V equals, pi r squared, times h\"></span>, where <span class=\"italic\">r</span> represents the radius of the base of the cylinder and <span class=\"italic\">h</span> represents the height. Since the height is 4 inches longer than the radius, the expression <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/da61d318c8c44b214220f38b6a67a09e399f61dfdc30aa1f5d8005728268c1d4\" alt=\"r plus 4 \"></span> represents the height of each cylindrical container. It follows that the volume of each container is represented by the equation <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/f6f1eae0a28ed469ba1c8e17f7d3a4bf3f1de22c440e27acd0a41a0fcd99993c\" alt=\"V equals, pi r squared times, open parenthesis, r plus 4, close parenthesis\"></span>. Distributing the expression <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/f4965677c6c69d7f8d902f877987923ee74d50df68cf1bc77c2d396245f47a4c\" alt=\"pi r squared \"></span> into each term in the parentheses yields <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/4493467ed86c96e7fdbe9a6ac5d4afa813dfe710b6e75070a2e7e169d3f69583\" alt=\"V equals, pi r cubed, plus 4 pi r squared\"></span>.</p><p>Choice A is incorrect and may result from representing the height as <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/b41772589f7bcb4563455d0651a9892835d3e1f226abca5404a75c7bf96892c3\" alt=\"4 r\"></span> instead of <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/da61d318c8c44b214220f38b6a67a09e399f61dfdc30aa1f5d8005728268c1d4\" alt=\"r plus 4\"></span>. Choice B is incorrect and may result from representing the height as <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/c457c0b10b5264872732faaedb8d56c861a656e4a05fcd2820a62b6deea52d5e\" alt=\"2 r\"></span> instead of <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/da61d318c8c44b214220f38b6a67a09e399f61dfdc30aa1f5d8005728268c1d4\" alt=\"r plus 4\"></span>. Choice C is incorrect and may result from representing the volume of a right cylinder as <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/c1a2e20ee250247ecb3ed5faee437f3592961a376fe894ae77b55897dca6a1ef\" alt=\"V equals pi r h\"></span> instead of <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/5a750e8de0d2c8dfee439ab17b14523361fdb26d14f609112027eee3efb10ed3\" alt=\"V equals, pi r squared, times h\"></span>.</p><p></p>\n","correct_answer":["D"],"has_media":true,"has_mathml":false,"has_table":false,"source_created_at":"2023-08-02T20:25:59.630000+00:00","source_updated_at":"2023-08-02T20:25:59.630000+00:00","options":[{"label":"A","content_html":"<p class=\"choice_paragraph \">\n            <span class=\"math_expression \">\n              <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/15fdefa168b8d94e097470769e7b257e9ac4433dbf79b081f7e5b2824a9b9265\" alt=\"V equals, 4 pi r cubed\"></span></span>\n          </p>\n","ord":0},{"label":"B","content_html":"<p class=\"choice_paragraph \">\n            <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/cad8799f29ffeafdb8457ac2e3c71185bfd184e4b096a8b22ee69c7ecf5634b4\" alt=\"V equals, pi times, open parenthesis, 2 r, close parenthesis, cubed\"></span></p>\n","ord":1},{"label":"C","content_html":"<p class=\"choice_paragraph \">\n            <span class=\"math_expression \">\n              <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/e7067bb1fce8ca276e3d799a9b38665f3b107216d013a355056f1c8231bd110d\" alt=\"V equals, pi r squared, plus 4 pi r\"></span></span>\n          </p>\n","ord":2},{"label":"D","content_html":"<p class=\"choice_paragraph \">\n            <span class=\"math_expression \">\n              <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/4493467ed86c96e7fdbe9a6ac5d4afa813dfe710b6e75070a2e7e169d3f69583\" alt=\"V equals, pi r cubed, plus 4 pi r squared\"></span></span>\n          </p>\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}}