{"id":"cb_question_bank:366ef2af-50e9-482e-9a5b-cf06361700e8","source_id":"cb_question_bank","source_item_id":"366ef2af-50e9-482e-9a5b-cf06361700e8","external_id":"4c9d9c55-af83-41a1-beaf-3cf5e261efd5","ibn":null,"question_id":"752055d1","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":"A scientist initially measures 12,000 bacteria in a growth medium. 4 hours later, the scientist measures 24,000 bacteria. Assuming exponential growth, the formula upper P equals upper C left parenthesis 2 right parenthesis Superscript r t gives the number of bacteria in the growth medium, where r and upper C are constants and upper P is the number of bacteria t hours after the initial measurement. What is the value of r ?","stimulus":"","stimulus_html":null,"stem_html":"<p>A scientist initially measures <math alttext=\"12,000\"><mn>12,000</mn>\n</math> bacteria in a growth medium. <math alttext=\"4\"><mn>4</mn>\n</math> hours later, the scientist measures <math alttext=\"24,000\"><mn>24,000</mn>\n</math> bacteria. Assuming exponential growth, the formula <math alttext=\"upper P equals upper C left parenthesis 2 right parenthesis Superscript r t\"><mi>P</mi><mo>=</mo><mi>C</mi><msup><mfenced><mn>2</mn></mfenced><mrow><mi>r</mi><mi>t</mi></mrow></msup></math> gives the number of bacteria in the growth medium, where <math alttext=\"r\"><mi>r</mi>\n</math> and <math alttext=\"upper C\"><mi>C</mi>\n</math> are constants and <math alttext=\"upper P\"><mi>P</mi>\n</math> is the number of bacteria <math alttext=\"t\"><mi>t</mi>\n</math><em>&nbsp;</em>hours after the initial measurement. What is the value of <math alttext=\"r\"><mi>r</mi>\n</math>?</p>","rationale_html":"<p>Choice B is correct. It’s given that the formula <math alttext=\"upper P equals upper C left parenthesis 2 right parenthesis Superscript r t\"><mi>P</mi><mo>=</mo><mi>C</mi><msup><mfenced><mn>2</mn></mfenced><mrow><mi>r</mi><mi>t</mi></mrow></msup></math> gives the number of bacteria in a growth medium, where <math alttext=\"r\"><mi>r</mi>\n</math> and <math alttext=\"upper C\"><mi>C</mi>\n</math> are constants and <math alttext=\"upper P\"><mi>P</mi>\n</math> is the number of bacteria <math alttext=\"t\"><mi>t</mi>\n</math> hours after the initial measurement. It’s also given that a scientist initially measures <math alttext=\"12,000\"><mn>12,000</mn>\n</math> bacteria in the growth medium. Since the initial measurement is <math alttext=\"0\"><mn>0</mn>\n</math> hours after the initial measurement, it follows that when <math alttext=\"t equals 0\"><mrow>\n\t<mi>t</mi>\n\t<mo>=</mo>\n\t<mn>0</mn>\n</mrow>\n</math>, <math alttext=\"upper P equals 12,000\"><mrow>\n\t<mi>P</mi>\n\t<mo>=</mo>\n\t<mn>12,000</mn>\n</mrow>\n</math>. Substituting <math alttext=\"0\"><mn>0</mn>\n</math> for <math alttext=\"t\"><mi>t</mi>\n</math> and <math alttext=\"12,000\"><mn>12,000</mn>\n</math> for <math alttext=\"upper P\"><mi>P</mi>\n</math> in the given equation yields <math alttext=\"12,000 equals upper C left parenthesis 2 right parenthesis Superscript r left parenthesis 0 right parenthesis\"><mn>12,000</mn><mo>=</mo><mi>C</mi><msup><mfenced><mn>2</mn></mfenced><mrow><mi>r</mi><mfenced><mn>0</mn></mfenced></mrow></msup></math>, or&nbsp;<math alttext=\"12,000 equals upper C left parenthesis 2 right parenthesis Superscript 0\"><mn>12,000</mn><mo>=</mo><mi>C</mi><msup><mfenced><mn>2</mn></mfenced><mn>0</mn></msup></math>, which is equivalent to <math alttext=\"12,000 equals upper C\"><mn>12,000</mn><mo>=</mo><mi>C</mi></math>. &nbsp;It’s given that <math alttext=\"4\"><mn>4</mn>\n</math> hours later, the scientist measures <math alttext=\"24,000\"><mn>24,000</mn>\n</math> bacteria, or when <math alttext=\"t equals 4\"><mrow>\n\t<mi>t</mi>\n\t<mo>=</mo>\n\t<mn>4</mn>\n</mrow>\n</math>, <math alttext=\"upper P equals 24,000\"><mrow>\n\t<mi>P</mi>\n\t<mo>=</mo>\n\t<mn>24,000</mn>\n</mrow>\n</math>. Substituting <math alttext=\"4\"><mn>4</mn>\n</math> for <math alttext=\"t\"><mi>t</mi>\n</math>, <math alttext=\"24,000\"><mn>24,000</mn>\n</math> for <math alttext=\"upper P\"><mi>P</mi>\n</math>, and <math alttext=\"12,000\"><mn>12,000</mn>\n</math> for <math alttext=\"upper C\"><mi>C</mi>\n</math> in the given equation yields <math alttext=\"24,000 equals 12,000 left parenthesis 2 right parenthesis Superscript 4 r\"><mn>24,000</mn><mo>=</mo><mn>12,000</mn><msup><mfenced><mn>2</mn></mfenced><mrow><mn>4</mn><mi>r</mi></mrow></msup></math>. Dividing each side of this equation by <math alttext=\"12,000\"><mn>12,000</mn>\n</math> yields <math alttext=\"2 equals 2 Superscript 4 r\"><mrow>\n\t<mn>2</mn>\n\t<mo>=</mo>\n\t<msup>\n\t\t<mn>2</mn>\n\t\t<mrow>\n\t\t\t<mn>4</mn>\n\t\t\t<mi>r</mi>\n\t\t</mrow>\n\t</msup>\n</mrow>\n</math>, or&nbsp;<math alttext=\"2 Superscript 1 Baseline equals 2 Superscript 4 r\"><msup><mn>2</mn><mn>1</mn></msup><mo>=</mo><msup><mn>2</mn><mrow><mn>4</mn><mi>r</mi></mrow></msup></math>, which is equivalent to <math alttext=\"1 equals 4 r\"><mrow>\n\t<mn>1</mn>\n\t<mo>=</mo>\n\t<mrow>\n\t\t<mn>4</mn>\n\t\t<mi>r</mi>\n\t</mrow>\n</mrow>\n</math>. Dividing both sides of this equation by <math alttext=\"4\"><mn>4</mn>\n</math> yields <math alttext=\"one fourth equals r\"><mrow>\n\t<mfrac>\n\t\t<mn>1</mn>\n\t\t<mn>4</mn>\n\t</mfrac>\n\t<mo>=</mo>\n\t<mi>r</mi>\n</mrow>\n</math>. Therefore, the value of <math alttext=\"r\"><mi>r</mi>\n</math> is <math alttext=\"one fourth\"><mfrac>\n\t<mn>1</mn>\n\t<mn>4</mn>\n</mfrac>\n</math>.</p>\n<p>Choice A is incorrect. This is the value of the reciprocal of <math alttext=\"upper C\"><mi>C</mi>\n</math>.</p>\n<p>Choice C is incorrect. This is the value of the reciprocal of <math alttext=\"r\"><mi>r</mi>\n</math>.</p>\n<p>Choice D is incorrect. This is the value of <math alttext=\"upper C\"><mi>C</mi>\n</math>.</p>","correct_answer":["B"],"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=\"StartFraction 1 Over 12,000 EndFraction\"><mfrac>\n\t<mn>1</mn>\n\t<mn>12,000</mn>\n</mfrac>\n</math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"one fourth\"><mfrac>\n\t<mn>1</mn>\n\t<mn>4</mn>\n</mfrac>\n</math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"4\"><mn>4</mn>\n</math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"12,000\"><mn>12,000</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}}