{"id":"cb_question_bank:0e9b8719-678b-4d1d-a21a-64060c2f6da9","source_id":"cb_question_bank","source_item_id":"0e9b8719-678b-4d1d-a21a-64060c2f6da9","external_id":null,"ibn":"029396-DC","question_id":"5afbdc8e","program":"SAT","module":"math","domain_code":"S","domain":"Geometry and Trigonometry","skill_code":"S.A.","skill":"Area and volume","difficulty":"M","score_band":5,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"What is the length of one side of a square that has the same area as a circle with radius 2 ?","stimulus":"","stimulus_html":null,"stem_html":"<p class=\"stem_paragraph \">What is the length of one side of a square that has the same area as a circle with radius 2 ?</p>\n","rationale_html":"<p>Choice C is correct. The area <span class=\"italic\">A</span> of a circle with radius <span class=\"italic\">r</span> is given by the formula <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/147a14a94af89b147131d66d162c9af8945b3984f9b6fc9a5b73e196485d7939\" alt=\"A equals, pi times r squared\"></span>. Thus, a circle with radius 2 has area <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/0367f5c977b143d3e9529db5bd8b2abc1e96f585d239b123443b331208456a1a\" alt=\"pi times 2 squared\"></span>, which can be rewritten as <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/69d50e47cc13180a2213f7d84c3bd4f5509b5aca75361d16b4eec507b921eef0\" alt=\"4 pi\"></span>. The area of a square with side length <span class=\"italic\">s</span> is given by the formula <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/f7fb2b4e7bdc57b8eb4726eafdcb69f749b8f218aff30c9747548e4ddafe5242\" alt=\"A equals s squared\"></span>. Thus, if a square has the same area as a circle with radius 2, then <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/3d5ad7b40c043d2441a73422123485e5a6d535278026791bb3d68c1d6f08f86a\" alt=\"s squared equals 4 pi\"></span>. Since the side length of a square must be a positive number, taking the square root of both sides of <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/3d5ad7b40c043d2441a73422123485e5a6d535278026791bb3d68c1d6f08f86a\" alt=\"s squared equals 4 pi\"></span> gives <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/b87ccb901754af019369b3462648b1ac45be87a31621bba484a8e2f51b619c83\" alt=\"s equals, the square root of 4 pi, end root\"></span>. Using the properties of square roots, <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/b486dbe27186199298b924f950d5f6138cd23ec27d281a5ce040b6f0c55bbc6e\" alt=\"the square root of 4 pi, end root \"></span> can be rewritten as <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/a137e770b82a0b99a5cb4eba1b03d5efa039cd47b1c4e8c4dea8b72afe407776\" alt=\"open parenthesis, the square root of 4, close parenthesis, times, open parenthesis, the square root of pi, close parenthesis\"></span>, which is equivalent to <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/b6ec9068d35c4b72046f33ede092d0fd030212fb5e77885a7f059db9297d256e\" alt=\"2 times the square root of pi\"></span>. Therefore, <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/f108f85d867113ea54fe057aeed8a96fe8aa254ffbf1411148f5a139988c71ca\" alt=\"s equals, 2 times the square root of pi\"></span>.</p><p>Choice A is incorrect. The side length of the square isn’t equal to the radius of the circle. Choices B and D are incorrect and may result from incorrectly simplifying the expression <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/b486dbe27186199298b924f950d5f6138cd23ec27d281a5ce040b6f0c55bbc6e\" alt=\"the square root of 4 pi, end root\"></span>.</p><p></p>\n","correct_answer":["C"],"has_media":true,"has_mathml":false,"has_table":false,"source_created_at":"2023-08-02T20:25:59.625000+00:00","source_updated_at":"2023-08-02T20:25:59.625000+00:00","options":[{"label":"A","content_html":"<p class=\"choice_paragraph \">2</p>\n","ord":0},{"label":"B","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/d0cf7b80bd131feb8251dc73256430e64d9feb86f30a475def19bda8d43fd618\" alt=\"the square root of 2 pi, end root\"></span></span>\n          </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/b6ec9068d35c4b72046f33ede092d0fd030212fb5e77885a7f059db9297d256e\" alt=\"2 times the square root of pi\"></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/a762bb431627bb855cc912defa0c579838a2b78cd709672e6adce83e8e91aac3\" alt=\"2 pi\"></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}}