{"id":"cb_question_bank:1f0e2444-ae3c-4f89-9187-756c4e090a5d","source_id":"cb_question_bank","source_item_id":"1f0e2444-ae3c-4f89-9187-756c4e090a5d","external_id":"01daee35-6a84-4baf-8ed5-3ef4ff55b251","ibn":null,"question_id":"9966235e","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 cube has an edge length of 68 inches. A solid sphere with a radius of 34 inches is inside the cube, such that the sphere touches the center of each face of the cube. To the nearest cubic inch, what is the volume of the space in the cube not taken up by the sphere?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">A cube has an edge length of <math alttext=\"68\"><mn>68</mn>\n</math> inches. A solid sphere with a radius of <math alttext=\"34\"><mn>34</mn>\n</math> inches is inside the cube, such that the sphere touches the center of each face of the cube. To the nearest cubic inch, what is the volume of the space in the cube <span style=\"text-decoration: underline;\">not</span> taken up by the sphere?</p>","rationale_html":"<p>Choice A is correct. The volume of a cube can be found by using the formula <math alttext=\"upper V equals s cubed\"><mi>V</mi><mo>=</mo><msup><mi>s</mi><mn>3</mn></msup></math>, where <math alttext=\"upper V\"><mi>V</mi>\n</math> is the volume and <math alttext=\"s\"><mi>s</mi>\n</math> is the edge length of the cube. Therefore, the volume of the given cube is <math alttext=\"upper V equals 68 cubed\"><mi>V</mi><mo>=</mo><msup><mn>68</mn><mn>3</mn></msup></math>, or <math alttext=\"314,432\"><mn>314,432</mn>\n</math> cubic inches. The volume of a sphere can be found by using the formula <math alttext=\"upper V equals four thirds pi r cubed\"><mi>V</mi><mo>=</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mi>π</mi><msup><mi>r</mi><mn>3</mn></msup></math> , where <math alttext=\"upper V\"><mi>V</mi>\n</math> is the volume and <math alttext=\"r\"><mi>r</mi>\n</math> is the radius of the sphere. Therefore, the volume of the given sphere is <math alttext=\"upper V equals four thirds pi left parenthesis 34 right parenthesis cubed\"><mi>V</mi><mo>=</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mi>π</mi><msup><mfenced><mn>34</mn></mfenced><mn>3</mn></msup></math>, or approximately <math alttext=\"164,636\"><mn>164,636</mn>\n</math> cubic inches. The volume of the space in the cube not taken up by the sphere is the difference between the volume of the cube and volume of the sphere. Subtracting the approximate volume of the sphere from the volume of the cube gives <math alttext=\"314,432 minus 164,636 equals 149,796\"><mn>314,432</mn><mo>-</mo><mn>164,636</mn><mo>=</mo><mn>149,796</mn></math>&nbsp;cubic inches.</p>\n<p>Choice B is incorrect and may result from conceptual or calculation errors.</p>\n<p>Choice C is incorrect and may result from conceptual or calculation errors.</p>\n<p>Choice D is incorrect and may result from conceptual or calculation errors.</p>","correct_answer":["A"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.836000+00:00","source_updated_at":"2023-08-02T20:25:59.836000+00:00","options":[{"label":"A","content_html":"<p><math alttext=\"149,796\"><mn>149,796</mn>\n</math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"164,500\"><mn>164,500</mn>\n</math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"190,955\"><mn>190,955</mn>\n</math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"310,800\"><mn>310,800</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}}