{"id":"cb_question_bank:ebb67cff-0be6-49f6-b0a8-95a8ca9ddcd1","source_id":"cb_question_bank","source_item_id":"ebb67cff-0be6-49f6-b0a8-95a8ca9ddcd1","external_id":"af32328c-4e46-4a53-b9c0-f627a4d608f9","ibn":null,"question_id":"7e5a3640","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.C.","skill":"Nonlinear functions","difficulty":"M","score_band":5,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"Bacteria are growing in a liquid growth medium. There were 300,000 cells per milliliter during an initial observation. The number of cells per milliliter doubles every 3 hours. How many cells per milliliter will there be 15 hours after the initial observation?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">Bacteria are growing in a liquid growth medium. There were <math alttext=\"300,000\"><mn>300,000</mn>\n</math> cells per milliliter during an initial observation. The number of cells per milliliter doubles every <math alttext=\"3\"><mn>3</mn>\n</math> hours. How many cells per milliliter will there be <math alttext=\"15\"><mn>15</mn>\n</math> hours after the initial observation?</p>","rationale_html":"<p style=\"text-align: left;\">Choice D is correct. Let <math alttext=\"y\"><mi>y</mi>\n</math> represent the number of cells per milliliter <math alttext=\"x\"><mi>x</mi>\n</math> hours after the initial observation. Since the number of cells per milliliter doubles every <math alttext=\"3\"><mn>3</mn>\n</math> hours, the relationship between <math alttext=\"x\"><mi>x</mi>\n</math> and <math alttext=\"y\"><mi>y</mi>\n</math> can be represented by an exponential equation of the form <math alttext=\"y equals a left parenthesis b right parenthesis Superscript StartFraction x Over k EndFraction\"><mi>y</mi><mo>=</mo><mi>a</mi><msup><mfenced><mi>b</mi></mfenced><mfrac><mi>x</mi><mi>k</mi></mfrac></msup></math>, where <math alttext=\"a\"><mi>a</mi>\n</math> is the number of cells per milliliter during the initial observation and the number of cells per milliliter increases by a factor of <math alttext=\"b\"><mi>b</mi>\n</math> every <math alttext=\"k\"><mi>k</mi>\n</math> hours. It’s given that there were <math alttext=\"300,000\"><mn>300,000</mn>\n</math> cells per milliliter during the initial observation. Therefore, <math alttext=\"a equals 300,000\"><mrow>\n\t<mi>a</mi>\n\t<mo>=</mo>\n\t<mn>300,000</mn>\n</mrow>\n</math>. It’s also given that the number of cells per milliliter doubles, or increases by a factor of <math alttext=\"2\"><mn>2</mn>\n</math>, every <math alttext=\"3\"><mn>3</mn>\n</math> hours. Therefore, <math alttext=\"b equals 2\"><mrow>\n\t<mi>b</mi>\n\t<mo>=</mo>\n\t<mn>2</mn>\n</mrow>\n</math> and <math alttext=\"k equals 3\"><mrow>\n\t<mi>k</mi>\n\t<mo>=</mo>\n\t<mn>3</mn>\n</mrow>\n</math>. Substituting <math alttext=\"300,000\"><mn>300,000</mn>\n</math> for <math alttext=\"a\"><mi>a</mi>\n</math>, <math alttext=\"2\"><mn>2</mn>\n</math> for <math alttext=\"b\"><mi>b</mi>\n</math>, and <math alttext=\"3\"><mn>3</mn>\n</math> for <math alttext=\"k\"><mi>k</mi>\n</math> in the equation <math alttext=\"y equals a left parenthesis b right parenthesis Superscript StartFraction x Over k EndFraction\"><mi>y</mi><mo>=</mo><mi>a</mi><msup><mfenced><mi>b</mi></mfenced><mfrac><mi>x</mi><mi>k</mi></mfrac></msup></math>&nbsp;yields <math alttext=\"y equals 300,000 left parenthesis 2 right parenthesis Superscript StartFraction x Over 3 EndFraction\"><mi>y</mi><mo>=</mo><mn>300,000</mn><msup><mfenced><mn>2</mn></mfenced><mfrac><mi>x</mi><mn>3</mn></mfrac></msup></math>. The number of cells per milliliter there will be <math alttext=\"15\"><mn>15</mn>\n</math> hours after the initial observation is the value of <math alttext=\"y\"><mi>y</mi>\n</math> in this equation when <math alttext=\"x equals 15\"><mrow>\n\t<mi>x</mi>\n\t<mo>=</mo>\n\t<mn>15</mn>\n</mrow>\n</math>. Substituting <math alttext=\"15\"><mn>15</mn>\n</math> for <math alttext=\"x\"><mi>x</mi>\n</math> in the equation <math alttext=\"y equals 300,000 left parenthesis 2 right parenthesis Superscript StartFraction x Over 3 EndFraction\"><mi>y</mi><mo>=</mo><mn>300,000</mn><msup><mfenced><mn>2</mn></mfenced><mfrac><mi>x</mi><mn>3</mn></mfrac></msup></math>&nbsp;yields <math alttext=\"y equals 300,000 left parenthesis 2 right parenthesis Superscript StartFraction 15 Over 3 EndFraction\"><mi>y</mi><mo>=</mo><mn>300,000</mn><msup><mfenced><mn>2</mn></mfenced><mfrac><mn>15</mn><mn>3</mn></mfrac></msup></math>, or <math alttext=\"y equals 300,000 left parenthesis 2 right parenthesis Superscript 5\"><mi>y</mi><mo>=</mo><mn>300,000</mn><msup><mfenced><mn>2</mn></mfenced><mn>5</mn></msup></math>. This is equivalent to&nbsp;<math alttext=\"y equals 300,000 left parenthesis 32 right parenthesis\"><mi>y</mi><mo>=</mo><mn>300,000</mn><mfenced><mn>32</mn></mfenced></math>, or <math alttext=\"y equals 9,600,000\"><mrow>\n\t<mi>y</mi>\n\t<mo>=</mo>\n\t<mn>9,600,000</mn>\n</mrow>\n</math>. Therefore, <math alttext=\"15\"><mn>15</mn>\n</math> hours after the initial observation, there will be <math alttext=\"9,600,000\"><mn>9,600,000</mn>\n</math> cells per milliliter.</p>\n<p style=\"text-align: left;\">Choice A is incorrect and may result from conceptual or calculation errors.</p>\n<p style=\"text-align: left;\">Choice B is incorrect&nbsp;and may result from conceptual or calculation errors.</p>\n<p style=\"text-align: left;\">Choice C is incorrect and may result from conceptual or calculation errors.</p>","correct_answer":["D"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.825000+00:00","source_updated_at":"2023-08-02T20:25:59.825000+00:00","options":[{"label":"A","content_html":"<p><math alttext=\"1,500,000\"><mn>1,500,000</mn>\n</math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"2,400,000\"><mn>2,400,000</mn>\n</math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"4,500,000\"><mn>4,500,000</mn>\n</math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"9,600,000\"><mn>9,600,000</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}}