{"id":"cb_question_bank:8aa8d0f8-e2ba-4482-ab49-7c1a6c5f5fe6","source_id":"cb_question_bank","source_item_id":"8aa8d0f8-e2ba-4482-ab49-7c1a6c5f5fe6","external_id":"e6248b96-6cdd-48a4-a20b-dc4b62d59d56","ibn":null,"question_id":"2d1614a1","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.C.","skill":"Nonlinear functions","difficulty":"H","score_band":6,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"upper P left parenthesis t right parenthesis equals 290 left parenthesis 1.04 right parenthesis Superscript left parenthesis four sixths right parenthesis t The function upper P models the population, in thousands, of a certain city t years after 2005 . According to the model, the population is predicted to increase by n percent sign every 18 months. What is the value of n ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: center;\"><math alttext=\"upper P left parenthesis t right parenthesis equals 290 left parenthesis 1.04 right parenthesis Superscript left parenthesis four sixths right parenthesis t\"><mi>P</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mrow><mn>290</mn></mrow><msup><mfenced><mrow><mn>1.04</mn></mrow></mfenced><mrow><mfenced><mfrac><mrow><mn>4</mn></mrow><mrow><mn>6</mn></mrow></mfrac></mfenced><mi>t</mi></mrow></msup></math></p>\n<p style=\"text-align: left;\">The function <math alttext=\"upper P\"><mi>P</mi>\n</math> models the population, in thousands, of a certain city <math alttext=\"t\"><mi>t</mi>\n</math> years after <math alttext=\"2005\"><mn>2005</mn>\n</math>. According to the model, the population is predicted to increase by&nbsp;<math alttext=\"n percent sign\"><mi>n</mi><mo>%</mo></math> every <math alttext=\"18\"><mn>18</mn>\n</math> months. What is the value of <math alttext=\"n\"><mi>n</mi>\n</math>?</p>","rationale_html":"<p>Choice C is correct. It's given that the function <math alttext=\"upper P\"><mi>P</mi>\n</math> models the population of the city <math alttext=\"t\"><mi>t</mi>\n</math> years after&nbsp;<math alttext=\"2005\"><mn>2005</mn></math>. Since there are <math alttext=\"12\"><mn>12</mn>\n</math> months in a year, <math alttext=\"18\"><mn>18</mn>\n</math> months is equivalent to <math alttext=\"StartFraction 18 Over 12 EndFraction\"><mfrac><mn>18</mn><mn>12</mn></mfrac></math> years. Therefore, the expression&nbsp;<math alttext=\"StartFraction 18 Over 12 EndFraction x\"><mfrac><mn>18</mn><mn>12</mn></mfrac><mi>x</mi></math> can represent the number of years in <math alttext=\"x\"><mi>x</mi>\n</math> <math alttext=\"18\"><mn>18</mn>\n</math>-month periods. Substituting&nbsp;<math alttext=\"StartFraction 18 Over 12 EndFraction x\"><mfrac><mn>18</mn><mn>12</mn></mfrac><mi>x</mi></math> for <math alttext=\"t\"><mi>t</mi>\n</math> in the given equation yields&nbsp;<math alttext=\"upper P left parenthesis StartFraction 18 Over 12 EndFraction x right parenthesis equals 290 left parenthesis 1.04 right parenthesis Superscript left parenthesis four sixths right parenthesis left parenthesis StartFraction 18 Over 12 EndFraction x right parenthesis\"><mi>P</mi><mfenced><mrow><mfrac><mn>18</mn><mn>12</mn></mfrac><mi>x</mi></mrow></mfenced><mo>=</mo><mn>290</mn><msup><mfenced><mn>1.04</mn></mfenced><mrow><mfenced><mfrac><mn>4</mn><mn>6</mn></mfrac></mfenced><mfenced><mrow><mfrac><mn>18</mn><mn>12</mn></mfrac><mi>x</mi></mrow></mfenced></mrow></msup></math>, which is equivalent to <math alttext=\"upper P left parenthesis StartFraction 18 Over 12 EndFraction x right parenthesis equals 290 left parenthesis 1.04 right parenthesis Superscript x\"><mi>P</mi><mfenced><mrow><mfrac><mn>18</mn><mn>12</mn></mfrac><mi>x</mi></mrow></mfenced><mo>=</mo><mn>290</mn><msup><mfenced><mn>1.04</mn></mfenced><mi>x</mi></msup></math>. Therefore, for each <math alttext=\"18\"><mn>18</mn>\n</math>-month period, the predicted population of the city is <math alttext=\"1.04\"><mn>1.04</mn>\n</math> times, or&nbsp;<math alttext=\"104 percent sign\"><mn>104</mn><mo>%</mo></math> of, the previous population. This means that the population is predicted to increase by&nbsp;<math alttext=\"4 percent sign\"><mn>4</mn><mo>%</mo></math> every <math alttext=\"18\"><mn>18</mn>\n</math> months.</p>\n<p>Choice A is incorrect and may result from conceptual or calculation errors.&nbsp;</p>\n<p>Choice B is incorrect. Each year, the predicted population of the city is <math alttext=\"1.04\"><mn>1.04</mn>\n</math> times the previous year's predicted population, which is not the same as an increase of <math alttext=\"1.04 percent sign\"><mn>1.04</mn><mo>%</mo></math>.</p>\n<p>Choice D is incorrect and may result from conceptual or calculation errors.</p>","correct_answer":["C"],"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":[{"label":"A","content_html":"<p><math alttext=\"0.38\"><mn>0.38</mn>\n</math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"1.04\"><mn>1.04</mn>\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=\"6\"><mn>6</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}}