{"id":"cb_question_bank:4d928223-f9f3-4cf7-8664-6c48aa62478b","source_id":"cb_question_bank","source_item_id":"4d928223-f9f3-4cf7-8664-6c48aa62478b","external_id":"7bb5425b-4812-479b-b73a-acf842444820","ibn":null,"question_id":"ce579859","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.C.","skill":"Nonlinear functions","difficulty":"H","score_band":7,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"A model estimates that at the end of each year from 2015 to 2020 , the number of squirrels in a population was 150 percent sign more than the number of squirrels in the population at the end of the previous year. The model estimates that at the end of 2016 , there were 180 squirrels in the population. Which of the following equations represents this model, where n is the estimated number of squirrels in the population t years after the end of 2015 and t less than or equals 5 ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">A model estimates that at the end of each year from&nbsp;<math alttext=\"2015\"><mn>2015</mn></math> to <math alttext=\"2020\"><mn>2020</mn></math>, the number of squirrels in a population was <math alttext=\"150 percent sign\"><mn>150</mn>\n<mo>%</mo></math> more than the number of squirrels in the population at the end of the previous year. The model estimates that at the end of <math alttext=\"2016\"><mn>2016</mn></math>, there were <math alttext=\"180\"><mn>180</mn>\n</math> squirrels in the population. Which of the following equations represents this model, where <math alttext=\"n\"><mi>n</mi>\n</math> is the estimated number of squirrels in the population <math alttext=\"t\"><mi>t</mi>\n</math> years after the end of <math alttext=\"2015\"><mn>2015</mn></math> and <math alttext=\"t less than or equals 5\"><mi>t</mi><mo>≤</mo><mn>5</mn></math>?</p>","rationale_html":"<p style=\"text-align: left;\">Choice B is correct. Since the model estimates that the number of squirrels in the population increased by a fixed percentage, <math alttext=\"150 percent sign\"><mn>150</mn><mo>%</mo></math>, each year, the model can be represented by an exponential equation of the form&nbsp;<math alttext=\"n equals a left parenthesis 1 plus StartFraction p Over 100 EndFraction right parenthesis Superscript t\"><mi>n</mi><mo>=</mo><mi>a</mi><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><mi>p</mi><mn>100</mn></mfrac></mrow></mfenced><mi>t</mi></msup></math>, where <math alttext=\"a\"><mi>a</mi>\n</math> is the estimated number of squirrels in the population at the end of&nbsp;<math alttext=\"2015\"><mn>2015</mn></math>, and the model estimates that at the end of each year, the number is <math alttext=\"p percent sign\"><mi>p</mi><mo>%</mo></math> more than the number at the end of the previous year. Since the model estimates that at the end of each year, the number was <math alttext=\"150 percent sign\"><mn>150</mn><mo>%</mo></math> more than the number at the end of the previous year, <math alttext=\"p equals 150\"><mrow>\n\t<mi>p</mi>\n\t<mo>=</mo>\n\t<mn>150</mn>\n</mrow>\n</math>. Substituting <math alttext=\"150\"><mn>150</mn>\n</math> for <math alttext=\"p\"><mi>p</mi>\n</math> in the equation <math alttext=\"n equals a left parenthesis 1 plus StartFraction p Over 100 EndFraction right parenthesis Superscript t\"><mi>n</mi><mo>=</mo><mi>a</mi><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><mi>p</mi><mn>100</mn></mfrac></mrow></mfenced><mi>t</mi></msup></math> yields <math alttext=\"n equals a left parenthesis 1 plus StartFraction 150 Over 100 EndFraction right parenthesis Superscript t\"><mi>n</mi><mo>=</mo><mi>a</mi><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><mn>150</mn><mn>100</mn></mfrac></mrow></mfenced><mi>t</mi></msup></math>, which is equivalent to <math alttext=\"n equals a left parenthesis 1 plus 1.5 right parenthesis Superscript t\"><mi>n</mi><mo>=</mo><mi>a</mi><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mn>1.5</mn></mrow></mfenced><mi>t</mi></msup></math>, or&nbsp;<math alttext=\"n equals a left parenthesis 2.5 right parenthesis Superscript t\"><mi>n</mi><mo>=</mo><mi>a</mi><msup><mfenced><mn>2.5</mn></mfenced><mi>t</mi></msup></math>. It’s given that the estimated number of squirrels at the end of <math alttext=\"2016\"><mn>2016</mn></math> was <math alttext=\"180\"><mn>180</mn>\n</math>. This means that when <math alttext=\"t equals 1\"><mrow>\n\t<mi>t</mi>\n\t<mo>=</mo>\n\t<mn>1</mn>\n</mrow>\n</math>, <math alttext=\"n equals 180\"><mrow>\n\t<mi>n</mi>\n\t<mo>=</mo>\n\t<mn>180</mn>\n</mrow>\n</math>. Substituting <math alttext=\"1\"><mn>1</mn>\n</math> for <math alttext=\"t\"><mi>t</mi>\n</math> and <math alttext=\"180\"><mn>180</mn>\n</math> for <math alttext=\"n\"><mi>n</mi>\n</math> in the equation&nbsp;<math alttext=\"n equals a left parenthesis 2.5 right parenthesis Superscript t\"><mi>n</mi><mo>=</mo><mi>a</mi><msup><mfenced><mn>2.5</mn></mfenced><mi>t</mi></msup></math> yields&nbsp;<math alttext=\"180 equals a left parenthesis 2.5 right parenthesis Superscript 1\"><mn>180</mn><mo>=</mo><mi>a</mi><msup><mfenced><mn>2.5</mn></mfenced><mn>1</mn></msup></math>, or <math alttext=\"180 equals 2.5 a\"><mrow>\n\t<mn>180</mn>\n\t<mo>=</mo>\n\t<mrow>\n\t\t<mn>2.5</mn>\n\t\t<mi>a</mi>\n\t</mrow>\n</mrow>\n</math>. Dividing each side of this equation by <math alttext=\"2.5\"><mn>2.5</mn>\n</math> yields <math alttext=\"72 equals a\"><mrow>\n\t<mn>72</mn>\n\t<mo>=</mo>\n\t<mi>a</mi>\n</mrow>\n</math>. Substituting <math alttext=\"72\"><mn>72</mn>\n</math> for <math alttext=\"a\"><mi>a</mi>\n</math> in the equation&nbsp;<math alttext=\"n equals a left parenthesis 2.5 right parenthesis Superscript t\"><mi>n</mi><mo>=</mo><mi>a</mi><msup><mfenced><mn>2.5</mn></mfenced><mi>t</mi></msup></math> yields&nbsp;<math alttext=\"n equals 72 left parenthesis 2.5 right parenthesis Superscript t\"><mi>n</mi><mo>=</mo><mn>72</mn><msup><mfenced><mn>2.5</mn></mfenced><mi>t</mi></msup></math>.</p>\n<p style=\"text-align: left;\">Choice A is incorrect. This equation represents a model where at the end of each year, the estimated number of squirrels was <math alttext=\"150 percent sign\"><mn>150</mn><mo>%</mo></math> of, not&nbsp;<math alttext=\"150 percent sign\"><mn>150</mn><mo>%</mo></math> more than, the estimated number at the end of the previous year.</p>\n<p style=\"text-align: left;\">Choice C is incorrect. This equation represents a model where at the end of each year, the estimated number of squirrels was <math alttext=\"150 percent sign\"><mn>150</mn><mo>%</mo></math>&nbsp;of, not&nbsp;<math alttext=\"150 percent sign\"><mn>150</mn><mo>%</mo></math> more than, the estimated number at the end of the previous year, and the estimated number of squirrels at the end of <math alttext=\"2015\"><mn>2015</mn></math>, not the end of&nbsp;<math alttext=\"2016\"><mn>2016</mn></math>, was <math alttext=\"180\"><mn>180</mn>\n</math>.</p>\n<p style=\"text-align: left;\">Choice D is incorrect. This equation represents a model where the estimated number of squirrels at the end of <math alttext=\"2015\"><mn>2015</mn></math>, not the end of&nbsp;<math alttext=\"2016\"><mn>2016</mn></math>, was <math alttext=\"180\"><mn>180</mn>\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=\"n equals 72 left parenthesis 1.5 right parenthesis Superscript t\"><mi>n</mi><mo>=</mo><mn>72</mn><msup><mfenced><mn>1.5</mn></mfenced><mi>t</mi></msup></math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"n equals 72 left parenthesis 2.5 right parenthesis Superscript t\"><mi>n</mi><mo>=</mo><mn>72</mn><msup><mfenced><mn>2.5</mn></mfenced><mi>t</mi></msup></math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"n equals 180 left parenthesis 1.5 right parenthesis Superscript t\"><mi>n</mi><mo>=</mo><mn>180</mn><msup><mfenced><mn>1.5</mn></mfenced><mi>t</mi></msup></math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"n equals 180 left parenthesis 2.5 right parenthesis Superscript t\"><mi>n</mi><mo>=</mo><mn>180</mn><msup><mfenced><mn>2.5</mn></mfenced><mi>t</mi></msup></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}}