{"id":"cb_question_bank:be6a42a0-ee10-474c-8a87-9e7b0e254d11","source_id":"cb_question_bank","source_item_id":"be6a42a0-ee10-474c-8a87-9e7b0e254d11","external_id":"edb987d4-772f-4ea6-8b75-8cd93a8fc5c8","ibn":null,"question_id":"d8ace155","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.C.","skill":"Nonlinear functions","difficulty":"M","score_band":4,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"A company opens an account with an initial balance of dollar sign 36,100.00 . The account earns interest, and no additional deposits or withdrawals are made. The account balance is given by an exponential function upper A , where upper A left parenthesis t right parenthesis is the account balance, in dollars, t years after the account is opened. The account balance after 13 years is dollar sign 68,071.93 . Which equation could define upper A ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">A company opens an account with an initial balance of <math alttext=\"dollar sign 36,100.00\"><mo>$</mo><mn>36,100.00</mn>\n</math>. The account earns interest, and no additional deposits or withdrawals are made. The account balance is given by an exponential function <math alttext=\"upper A\"><mi>A</mi>\n</math>, where <math alttext=\"upper A left parenthesis t right parenthesis\"><mi>A</mi><mo>(</mo><mi>t</mi><mo>)</mo></math> is the account balance, in dollars, <math alttext=\"t\"><mi>t</mi>\n</math> years after the account is opened. The account balance after <math alttext=\"13\"><mn>13</mn>\n</math> years is <math alttext=\"dollar sign 68,071.93\"><mo>$</mo><mn>68,071.93</mn>\n</math>. Which equation could define <math alttext=\"upper A\"><mi>A</mi>\n</math>?</p>","rationale_html":"<p>Choice A is correct. Since it's given that the account balance, <math alttext=\"upper A left parenthesis t right parenthesis\"><mi>A</mi><mfenced><mi>t</mi></mfenced></math>, in dollars, after <math alttext=\"t\"><mi>t</mi>\n</math> years can be modeled by an exponential function, it follows that function <math alttext=\"upper A\"><mi>A</mi>\n</math> can be written in the form <math alttext=\"upper A left parenthesis t right parenthesis equals upper N r Superscript t\"><mi>A</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mi>N</mi><msup><mi>r</mi><mi>t</mi></msup></math>, where <math alttext=\"upper N\"><mi>N</mi>\n</math> is the initial value of the function and <math alttext=\"r\"><mi>r</mi>\n</math> is a constant related to the growth of the function. It's given that the initial balance of the account is <math alttext=\"dollar sign 36,100.00\"><mo>$</mo><mn>36,100.00</mn></math>, so it follows that the initial value of the function, or <math alttext=\"upper N\"><mi>N</mi>\n</math>, must be&nbsp;<math alttext=\"36,100.00\"><mn>36,100.00</mn></math>. Substituting&nbsp;<math alttext=\"36,100.00\"><mn>36,100.00</mn></math> for <math alttext=\"upper N\"><mi>N</mi>\n</math> in the equation&nbsp;<math alttext=\"upper A left parenthesis t right parenthesis equals upper N r Superscript t\"><mi>A</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mi>N</mi><msup><mi>r</mi><mi>t</mi></msup></math> yields&nbsp;<math alttext=\"upper A left parenthesis t right parenthesis equals 36,100.00 r Superscript t\"><mi>A</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mn>36,100.00</mn><msup><mi>r</mi><mi>t</mi></msup></math>. It's given that the account balance after <math alttext=\"13\"><mn>13</mn>\n</math> years, or when <math alttext=\"t equals 13\"><mi>t</mi><mo>=</mo><mn>13</mn></math>, is&nbsp;<math alttext=\"dollar sign 68,071.93\"><mo>$</mo><mn>68,071.93</mn></math>. It follows that <math alttext=\"upper A left parenthesis 13 right parenthesis equals 68,071.93\"><mi>A</mi><mfenced><mn>13</mn></mfenced><mo>=</mo><mn>68,071.93</mn></math>, or <math alttext=\"36,100.00 r Superscript 13 Baseline equals 68,071.93\"><mn>36,100.00</mn><msup><mi>r</mi><mn>13</mn></msup><mo>=</mo><mn>68,071.93</mn></math>. Dividing each side of the equation&nbsp;<math alttext=\"36,100.00 r Superscript 13 Baseline equals 68,071.93\"><mn>36,100.00</mn><msup><mi>r</mi><mn>13</mn></msup><mo>=</mo><mn>68,071.93</mn></math> by&nbsp;<math alttext=\"36,100.00\"><mn>36,100.00</mn></math> yields&nbsp;<math alttext=\"r Superscript 13 Baseline equals StartFraction 68,071.93 Over 36,100.00 EndFraction\"><msup><mi>r</mi><mn>13</mn></msup><mo>=</mo><mfrac><mn>68,071.93</mn><mn>36,100.00</mn></mfrac></math>. Taking the <math alttext=\"13\"><mn>13</mn>\n</math>th root of both sides of this equation yields <math alttext=\"r equals RootIndex 13 StartRoot StartFraction 68,071.93 Over 36,100.00 EndFraction EndRoot\"><mi>r</mi><mo>=</mo><mroot><mfrac><mn>68,071.93</mn><mn>36,100.00</mn></mfrac><mn>13</mn></mroot></math>, or <math alttext=\"r\"><mi>r</mi>\n</math> is approximately equal to <math alttext=\"1.05\"><mn>1.05</mn>\n</math>. Substituting <math alttext=\"1.05\"><mn>1.05</mn>\n</math> for <math alttext=\"r\"><mi>r</mi>\n</math> in the equation <math alttext=\"upper A left parenthesis t right parenthesis equals 36,100.00 r Superscript t\"><mi>A</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mn>36,100.00</mn><msup><mi>r</mi><mi>t</mi></msup></math> yields&nbsp;<math alttext=\"upper A left parenthesis t right parenthesis equals 36,100.00 left parenthesis 1.05 right parenthesis Superscript t\"><mi>A</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mn>36,100.00</mn><msup><mfenced><mn>1.05</mn></mfenced><mi>t</mi></msup></math>, so the equation <math alttext=\"upper A left parenthesis t right parenthesis equals 36,100.00 left parenthesis 1.05 right parenthesis Superscript t\"><mi>A</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mn>36,100.00</mn><msup><mfenced><mn>1.05</mn></mfenced><mi>t</mi></msup></math> could define <math alttext=\"upper A\"><mi>A</mi>\n</math>.&nbsp;</p>\n<p>Choice B is incorrect. Substituting <math alttext=\"0\"><mn>0</mn>\n</math> for <math alttext=\"t\"><mi>t</mi>\n</math> in this function indicates an initial balance of <math alttext=\"dollar sign 31,971.93\"><mo>$</mo><mn>31,971.93</mn></math>, rather than&nbsp;<math alttext=\"dollar sign 36,100.00\"><mo>$</mo><mn>36,100.00</mn></math>.</p>\n<p>Choice C is incorrect. Substituting <math alttext=\"0\"><mn>0</mn>\n</math> for <math alttext=\"t\"><mi>t</mi>\n</math> in this function indicates an initial balance of <math alttext=\"dollar sign 31,971.93\"><mo>$</mo><mn>31,971.93</mn></math>, rather than&nbsp;<math alttext=\"dollar sign 36,100.00\"><mo>$</mo><mn>36,100.00</mn></math>. Additionally, this function indicates the account balance is decreasing, rather than increasing, over time.&nbsp;</p>\n<p>Choice D is incorrect. This function indicates the account balance is decreasing, rather than increasing, over time.</p>","correct_answer":["A"],"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=\"upper A left parenthesis t right parenthesis equals 36,100.00 left parenthesis 1.05 right parenthesis Superscript t\"><mi>A</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mn>36,100.00</mn>\n<msup><mfenced><mrow><mn>1.05</mn></mrow></mfenced><mi>t</mi></msup></math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"upper A left parenthesis t right parenthesis equals 31,971.93 left parenthesis 1.05 right parenthesis Superscript t\"><mi>A</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mn>31,971.93</mn>\n<msup><mfenced><mrow><mn>1.05</mn></mrow></mfenced><mi>t</mi></msup></math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"upper A left parenthesis t right parenthesis equals 31,971.93 left parenthesis 0.05 right parenthesis Superscript t\"><mi>A</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mn>31,971.93</mn>\n<msup><mfenced><mrow><mn>0.05</mn></mrow></mfenced><mi>t</mi></msup></math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"upper A left parenthesis t right parenthesis equals 36,100.00 left parenthesis 0.05 right parenthesis Superscript t\"><mi>A</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mn>36,100.00</mn>\n<msup><mfenced><mrow><mn>0.05</mn></mrow></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}}