{"id":"cb_question_bank:8936ba3a-68ed-4a79-b894-5d790adff80e","source_id":"cb_question_bank","source_item_id":"8936ba3a-68ed-4a79-b894-5d790adff80e","external_id":null,"ibn":"032655-DC","question_id":"dc71597b","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 right circular cone has a volume of one third, pi cubic feet and a height of 9 feet. What is the radius, in feet, of the base of the cone?","stimulus":"","stimulus_html":null,"stem_html":"<p class=\"stem_paragraph \">A right circular cone has a volume of <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/388e2f1b864a9cab0c15b3071d37166581a789e7248737078b09e8237d8adca1\" alt=\"one third, pi \"></span></span>&nbsp;cubic feet and a height of 9&nbsp;feet. What is the radius, in feet, of the base of the cone?</p>\n","rationale_html":"<p>Choice A is correct. The equation for the volume of a right circular cone is <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/cb158265b2920cd5de4712a1885e2dd39fb1f5298f0859d6ca71ddecd67c0421\" alt=\"V equals, one third pi r squared times h\"></span>. It’s given that the volume of the right circular cone is <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/388e2f1b864a9cab0c15b3071d37166581a789e7248737078b09e8237d8adca1\" alt=\"one third pi \"></span> cubic feet and the height is 9 feet. Substituting these values for <span class=\"italic\">V</span> and <span class=\"italic\">h</span>, respectively, gives <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/eb41ea1ec32a55cd6af9e834dd67aef57a61c601a85b773001cc2411795f9658\" alt=\"one third pi equals, one third pi r squared, times 9\"></span>. Dividing both sides of the equation by <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/388e2f1b864a9cab0c15b3071d37166581a789e7248737078b09e8237d8adca1\" alt=\"one third pi \"></span> gives <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/bf0d9bfce82dd135c24a587fd5eb3e02d6c11bcf5a096a1b3ee8931b7a26ddae\" alt=\"1 equals, r squared times 9\"></span>. Dividing both sides of the equation by 9 gives <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/dc57bffdbdc26ddc0d24cfca151a61376fcd0cc557c53e3124fedb5efe2582a3\" alt=\"one ninth equals r squared\"></span>. Taking the square root of both sides results in two possible values for the radius, <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/0283196cc4d93ceb562022789cff50ccf910db53fac357df937d530dd5d98e93\" alt=\"the square root of one ninth \"></span> or <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/0318ae19e4898632f45fb5c46fe7363260ca72f37b54e3423acc64838f87598b\" alt=\"the negative of the square root of one ninth\"></span>. Since the radius can’t have a negative value, that leaves <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/0283196cc4d93ceb562022789cff50ccf910db53fac357df937d530dd5d98e93\" alt=\"the square root of one ninth \"></span> as the only possibility. Applying the quotient property of square roots, <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/13b57770127a35c49688ae22ee7f594e3b24ed7240259b26a7b5db7210fe5404\" alt=\"the square root of the fraction a, over b, equals, the fraction the square root of a, over the square root of b\"></span>, results in <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/58c23bea9db502d68a3bfe3bb37811fecb230eabc2370ac8689cdb22e2028f0b\" alt=\"r equals, the fraction the square root of 1 over the square root of 9\"></span>, or <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/4dec0de2022f970fb92ed1a116d0c6a064cd8492eb784592db37a54f0235e008\" alt=\"r equals one third\"></span>.</p><p>Choices B and C are incorrect and may result from incorrectly evaluating <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/0283196cc4d93ceb562022789cff50ccf910db53fac357df937d530dd5d98e93\" alt=\"the square root of one ninth\"></span>. Choice D is incorrect and may result from solving <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/f90d72715ad0202a8ca81e41240b3131fbae9e3a4da8db3bcd1cfce14cf071c3\" alt=\"r squared equals 9 \"></span> instead of <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/740f123cb0a87a1486b70c1d524da733d4892a4cdc5a3bacfbeb47f3ced78d15\" alt=\"r squared equals one ninth\"></span>.</p><p></p>\n","correct_answer":["A"],"has_media":true,"has_mathml":false,"has_table":false,"source_created_at":"2023-08-02T20:25:59.627000+00:00","source_updated_at":"2023-08-02T20:25:59.627000+00:00","options":[{"label":"A","content_html":"<span class=\"choice_cell align:center \">\n            <span class=\"math_expression \">\n              <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/fce26b7f770543c2e33190a4ea311aa1f4d8da39df9b43ef0ea5dfff882a8571\" alt=\"one third\"></span></span>\n          </span>\n","ord":0},{"label":"B","content_html":"<span class=\"choice_cell align:center \">\n            <span class=\"math_expression \">\n              <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/d00c3eca46ba1ac866a5491aba0100067b577055fe1ab2b9286d215af7319a80\" alt=\"the fraction 1 over the square root of 3, end fraction\"></span></span>\n          </span>\n","ord":1},{"label":"C","content_html":"<span class=\"choice_cell align:center \">\n            <span class=\"math_expression \">\n              <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/548f291932b05f462911ef8ecc28c868d5c8cd912ea251c39ba1e00283050296\" alt=\"the square root of 3\"></span></span>\n          </span>\n","ord":2},{"label":"D","content_html":"<span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/b35ae18904fb781bd47e663cf02b992965491e4f50b31194e7d69870143879cd\" alt=\"3\"></span>\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}}