{"id":"cb_question_bank:c6ca457a-497f-45a8-847f-fd110769f249","source_id":"cb_question_bank","source_item_id":"c6ca457a-497f-45a8-847f-fd110769f249","external_id":"64a7c0bf-065d-47a5-a938-4a1fabad6933","ibn":null,"question_id":"f329442c","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":"Circle upper A has a radius of 3 n and circle upper B has a radius of 129 n , where n is a positive constant. The area of circle upper B is how many times the area of circle upper A ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">Circle <math alttext=\"upper A\"><mi>A</mi>\n</math><em>&nbsp;</em>has a radius of <math alttext=\"3 n\"><mrow>\n\t<mn>3</mn>\n\t<mi>n</mi>\n</mrow>\n</math> and circle <math alttext=\"upper B\"><mi>B</mi>\n</math><em>&nbsp;</em>has a radius of <math alttext=\"129 n\"><mrow>\n\t<mn>129</mn>\n\t<mi>n</mi>\n</mrow>\n</math>, where <math alttext=\"n\"><mi>n</mi>\n</math> is a positive constant. The area of circle <math alttext=\"upper B\"><mi>B</mi>\n</math><em>&nbsp;</em>is how many times the area of circle <math alttext=\"upper A\"><mi>A</mi>\n</math>?</p>","rationale_html":"<p>Choice D is correct. The area of a circle can be found by using the formula <math alttext=\"upper A equals pi r squared\"><mi>A</mi><mo>=</mo><mi>π</mi><msup><mi>r</mi><mn>2</mn></msup></math>, where <math alttext=\"upper A\"><mi>A</mi>\n</math> is the area and <math alttext=\"r\"><mi>r</mi>\n</math> is the radius of the circle. It’s given that the radius of circle <em>A</em> is <math alttext=\"3 n\"><mn>3</mn><mi>n</mi></math>. Substituting this value for <math alttext=\"r\"><mi>r</mi>\n</math> into the formula&nbsp;<math alttext=\"upper A equals pi r squared\"><mi>A</mi><mo>=</mo><mi>π</mi><msup><mi>r</mi><mn>2</mn></msup></math> gives <math alttext=\"upper A equals pi left parenthesis 3 n right parenthesis squared\"><mi>A</mi><mo>=</mo><mi>π</mi><msup><mfenced><mrow><mn>3</mn><mi>n</mi></mrow></mfenced><mn>2</mn></msup></math>, or <math alttext=\"9 pi n squared\"><mn>9</mn><mi>π</mi><msup><mi>n</mi><mn>2</mn></msup></math>. It’s also given that the radius of circle <em>B</em> is <math alttext=\"129 n\"><mn>129</mn><mi>n</mi></math>. Substituting this value for <math alttext=\"r\"><mi>r</mi>\n</math> into the formula <math alttext=\"upper A equals pi r squared\"><mi>A</mi><mo>=</mo><mi>π</mi><msup><mi>r</mi><mn>2</mn></msup></math> gives <math alttext=\"upper A equals pi left parenthesis 129 n right parenthesis squared\"><mi>A</mi><mo>=</mo><mi>π</mi><msup><mfenced><mrow><mn>129</mn><mi>n</mi></mrow></mfenced><mn>2</mn></msup></math>, or <math alttext=\"16,641 pi n squared\"><mn>16,641</mn><mi>π</mi><msup><mi>n</mi><mn>2</mn></msup></math>. Dividing the area of circle <em>B</em> by the area of circle <em>A</em> gives <math alttext=\"StartFraction 16,641 pi n squared Over 9 pi n squared EndFraction\"><mfrac><mrow><mn>16,641</mn><mi>π</mi><msup><mi>n</mi><mn>2</mn></msup></mrow><mrow><mn>9</mn><mi>π</mi><msup><mi>n</mi><mn>2</mn></msup></mrow></mfrac></math>, which simplifies to <math alttext=\"1,849\"><mn>1,849</mn>\n</math>. Therefore, the area of circle <em>B</em> is <math alttext=\"1,849\"><mn>1,849</mn>\n</math> times the area of circle <em>A</em>.</p>\n<p>Choice A is incorrect. This is how many times greater the radius of circle <em>B</em> is than the radius of circle <em>A</em>.</p>\n<p>Choice B is incorrect and may result from conceptual or calculation errors.</p>\n<p>Choice C is incorrect. This is the coefficient on the term that describes the radius of circle <em>B</em>.</p>","correct_answer":["D"],"has_media":false,"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":"<p><math alttext=\"43\"><mn>43</mn>\n</math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"86\"><mn>86</mn>\n</math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"129\"><mn>129</mn>\n</math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"1,849\"><mn>1,849</mn>\n</math></p>","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}}