{"id":"cb_question_bank:8530c2c6-f4d8-4b1a-85d0-395d5d4fa553","source_id":"cb_question_bank","source_item_id":"8530c2c6-f4d8-4b1a-85d0-395d5d4fa553","external_id":"6951ac71-d64f-4a32-b84a-46f30f206692","ibn":null,"question_id":"90bcaa61","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.C.","skill":"Nonlinear functions","difficulty":"M","score_band":5,"answer_type":"spr","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"The function f left parenthesis t right parenthesis equals 60,000 left parenthesis 2 right parenthesis Superscript StartFraction t Over 410 EndFraction gives the number of bacteria in a population t minutes after an initial observation. How much time, in minutes, does it take for the number of bacteria in the population to double?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">The function&nbsp;<math alttext=\"f left parenthesis t right parenthesis equals 60,000 left parenthesis 2 right parenthesis Superscript StartFraction t Over 410 EndFraction\"><mi>f</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mrow><mn>60,000</mn></mrow><msup><mfenced><mn>2</mn></mfenced><mfrac><mi>t</mi><mrow><mn>410</mn></mrow></mfrac></msup></math> gives the number of bacteria in a population <math alttext=\"t\"><mi>t</mi>\n</math> minutes after an initial observation. How much time, in minutes, does it take for the number of bacteria in the population to double?</p>","rationale_html":"<p style=\"text-align: left;\">The correct answer is <math alttext=\"410\"><mn>410</mn>\n</math>. It's given that <math alttext=\"t\"><mi>t</mi>\n</math> minutes after an initial observation, the number of bacteria in a population is <math alttext=\"60,000 left parenthesis 2 right parenthesis Superscript StartFraction t Over 410 EndFraction\"><mn>60,000</mn><msup><mfenced><mn>2</mn></mfenced><mfrac><mi>t</mi><mn>410</mn></mfrac></msup></math>. This expression consists of the initial number of bacteria, <math alttext=\"60,000\"><mn>60,000</mn></math>, multiplied by the expression <math alttext=\"2 Superscript StartFraction t Over 410 EndFraction\"><msup><mn>2</mn><mfrac><mi>t</mi><mn>410</mn></mfrac></msup></math>. The time it takes for the number of bacteria to double is the increase in the value of <math alttext=\"t\"><mi>t</mi>\n</math> that causes the expression <math alttext=\"2 Superscript StartFraction t Over 410 EndFraction\"><msup><mn>2</mn><mfrac><mi>t</mi><mn>410</mn></mfrac></msup></math> to double. Since the base of the expression&nbsp;<math alttext=\"2 Superscript StartFraction t Over 410 EndFraction\"><msup><mn>2</mn><mfrac><mi>t</mi><mn>410</mn></mfrac></msup></math> is <math alttext=\"2\"><mn>2</mn>\n</math>, the expression <math alttext=\"2 Superscript StartFraction t Over 410 EndFraction\"><msup><mn>2</mn><mfrac><mi>t</mi><mn>410</mn></mfrac></msup></math> will double when the exponent increases by <math alttext=\"1\"><mn>1</mn>\n</math>. Since the exponent of the expression&nbsp;<math alttext=\"2 Superscript StartFraction t Over 410 EndFraction\"><msup><mn>2</mn><mfrac><mi>t</mi><mn>410</mn></mfrac></msup></math> is <math alttext=\"StartFraction t Over 410 EndFraction\"><mfrac><mi>t</mi><mn>410</mn></mfrac></math>, the exponent will increase by <math alttext=\"1\"><mn>1</mn>\n</math> when <math alttext=\"t\"><mi>t</mi>\n</math> increases by <math alttext=\"410\"><mn>410</mn>\n</math>. Therefore the time, in minutes, it takes for the number of bacteria in the population to double is <math alttext=\"410\"><mn>410</mn>\n</math>.</p>","correct_answer":["410"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.826000+00:00","source_updated_at":"2023-08-02T20:25:59.826000+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}}