{"id":"cb_question_bank:4e6f07b5-a8ed-4de9-9a9a-5448ce0c4f2f","source_id":"cb_question_bank","source_item_id":"4e6f07b5-a8ed-4de9-9a9a-5448ce0c4f2f","external_id":"243579cd-f8bb-4573-bce8-ea57314f3f36","ibn":null,"question_id":"899c6042","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":"spr","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"A right circular cone has a height of 22 centimeters left parenthesis cm right parenthesis and a base with a diameter of 6 cm . The volume of this cone is n pi cm cubed . What is the value of n ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">A right circular cone has a height of <math alttext=\"22 centimeters left parenthesis cm right parenthesis\"><mrow><mn>22</mn></mrow><mo>&nbsp;</mo><mtext>centimeters</mtext><mo>&nbsp;</mo><mfenced><mtext>cm</mtext></mfenced></math> and a base with a diameter of <math alttext=\"6 cm\"><mrow><mn>6</mn></mrow><mo>&nbsp;</mo><mtext>cm</mtext></math>. The volume of this cone is <math alttext=\"n pi cm cubed\"><mi>n</mi><mi>π</mi><mo>&nbsp;</mo><msup><mtext>cm</mtext><mn>3</mn></msup></math>. What is the value of <math alttext=\"n\"><mi>n</mi>\n</math>?</p>","rationale_html":"<p style=\"text-align: left;\">The correct answer is <math alttext=\"66\"><mn>66</mn>\n</math>. It’s given that the right circular cone has a height of <math alttext=\"22\"><mn>22</mn>\n</math> centimeters <math alttext=\"left parenthesis cm right parenthesis\"><mfenced><mtext>cm</mtext></mfenced></math> and a base with a diameter of <math alttext=\"6 cm\"><mn>6</mn><mo>&nbsp;</mo><mtext>cm</mtext></math>. Since the diameter of the base of the cone is <math alttext=\"6 cm\"><mn>6</mn><mo>&nbsp;</mo><mtext>cm</mtext></math>, the radius of the base is <math alttext=\"3 cm\"><mn>3</mn><mo>&nbsp;</mo><mtext>cm</mtext></math>. The volume <math alttext=\"upper V\"><mi>V</mi>\n</math>, <math alttext=\"in cm cubed\"><mtext>in</mtext><mo>&nbsp;</mo><msup><mtext>cm</mtext><mn>3</mn></msup></math>, of a right circular cone can be found using the formula <math alttext=\"upper V equals one third pi r squared h\"><mi>V</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>π</mi><msup><mi>r</mi><mn>2</mn></msup><mi>h</mi></math>, where <math alttext=\"h\"><mi>h</mi>\n</math> is the height,&nbsp;<math alttext=\"in cm\"><mtext>in</mtext><mo>&nbsp;</mo><mtext>cm</mtext></math>, and <math alttext=\"r\"><mi>r</mi>\n</math> is the radius,&nbsp;<math alttext=\"in cm\"><mtext>in</mtext><mo>&nbsp;</mo><mtext>cm</mtext></math>,&nbsp;of the base of the cone. Substituting <math alttext=\"22\"><mn>22</mn>\n</math> for <math alttext=\"h\"><mi>h</mi>\n</math> and <math alttext=\"3\"><mn>3</mn>\n</math> for <math alttext=\"r\"><mi>r</mi>\n</math> in this formula yields <math alttext=\"upper V equals one third pi left parenthesis 3 right parenthesis squared left parenthesis 22 right parenthesis\"><mi>V</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>π</mi><msup><mfenced><mn>3</mn></mfenced><mn>2</mn></msup><mfenced><mn>22</mn></mfenced></math>, or <math alttext=\"upper V equals 66 pi\"><mi>V</mi><mo>=</mo><mn>66</mn><mi>π</mi></math>. Therefore, the volume of the cone is <math alttext=\"66 pi italic cm cubed\"><mn>66</mn><mi>π</mi><mo>&nbsp;</mo><msup><mtext>cm</mtext><mn>3</mn></msup></math>. It’s given that the volume of the cone is <math alttext=\"n pi italic cm cubed\"><mi>n</mi><mi>π</mi><mo>&nbsp;</mo><msup><mtext>cm</mtext><mn>3</mn></msup></math>. Therefore, the value of <math alttext=\"n\"><mi>n</mi>\n</math> is <math alttext=\"66\"><mn>66</mn>\n</math>.</p>","correct_answer":["66"],"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":[],"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}}