{"id":"cb_question_bank:12382f2f-1a98-4789-8a6f-be114f1ed35f","source_id":"cb_question_bank","source_item_id":"12382f2f-1a98-4789-8a6f-be114f1ed35f","external_id":"e78c85a7-41e7-485b-b784-99727df47183","ibn":null,"question_id":"52b1700c","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":"Time (years) Total amount (dollars) 0 604.00 1 606.42 2 608.84 Rosa opened a savings account at a bank. The table shows the exponential relationship between the time t , in years, since Rosa opened the account and the total amount n , in dollars, in the account. If Rosa made no additional deposits or withdrawals, which of the following equations best represents the relationship between t and n ?","stimulus":"","stimulus_html":null,"stem_html":"<table style=\"border-collapse: collapse; width: 100%; border-color: #000000; border-style: solid; margin-left: auto; margin-right: auto;\" border=\"1\">\n<thead>\n<tr>\n<th style=\"width: 47.1852%; text-align: center;\" scope=\"col\">Time (years)</th>\n<th style=\"width: 47.3594%; text-align: center;\" scope=\"col\">Total amount (dollars)</th>\n</tr>\n</thead>\n<tbody>\n<tr>\n<td style=\"width: 47.1852%; text-align: center;\"><math alttext=\"0\"><mn>0</mn></math></td>\n<td style=\"width: 47.3594%; text-align: center;\"><math alttext=\"604.00\"><mn>604.00</mn>\n</math></td>\n</tr>\n<tr>\n<td style=\"width: 47.1852%; text-align: center;\"><math alttext=\"1\"><mn>1</mn></math></td>\n<td style=\"width: 47.3594%; text-align: center;\"><math alttext=\"606.42\"><mn>606.42</mn>\n</math></td>\n</tr>\n<tr>\n<td style=\"width: 47.1852%; text-align: center;\"><math alttext=\"2\"><mn>2</mn></math></td>\n<td style=\"width: 47.3594%; text-align: center;\"><math alttext=\"608.84\"><mn>608.84</mn>\n</math></td>\n</tr>\n</tbody>\n</table>\n<p style=\"text-align: left;\">Rosa opened a savings account at a bank. The table shows the exponential relationship between the time <math alttext=\"t\"><mi>t</mi>\n</math>, in years, since Rosa opened the account and the total amount <math alttext=\"n\"><mi>n</mi>\n</math>, in dollars, in the account. If Rosa made no additional deposits or withdrawals, which of the following equations best represents the relationship between <math alttext=\"t\"><mi>t</mi>\n</math> and <math alttext=\"n\"><mi>n</mi>\n</math>?</p>","rationale_html":"<p style=\"text-align: left;\">Choice C is correct. It’s given that the relationship between <math alttext=\"t\"><mi>t</mi>\n</math> and <math alttext=\"n\"><mi>n</mi>\n</math> is exponential. The table shows that the value of <math alttext=\"n\"><mi>n</mi>\n</math> increases as the value of <math alttext=\"t\"><mi>t</mi>\n</math> increases. Therefore, the relationship between <math alttext=\"t\"><mi>t</mi>\n</math> and <math alttext=\"n\"><mi>n</mi>\n</math> can be represented by an increasing exponential equation of the form <math alttext=\"n equals a left parenthesis 1 plus b right parenthesis Superscript t\"><mi>n</mi><mo>=</mo><mi>a</mi><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>b</mi></mrow></mfenced><mi>t</mi></msup></math>, where <math alttext=\"a\"><mi>a</mi>\n</math> and <math alttext=\"b\"><mi>b</mi>\n</math> are positive constants. The table shows 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=\"n equals 604\"><mrow>\n\t<mi>n</mi>\n\t<mo>=</mo>\n\t<mn>604</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=\"604\"><mn>604</mn>\n</math> for <math alttext=\"n\"><mi>n</mi>\n</math> in the equation <math alttext=\"n equals a left parenthesis 1 plus b right parenthesis Superscript t\"><mi>n</mi><mo>=</mo><mi>a</mi><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>b</mi></mrow></mfenced><mi>t</mi></msup></math> yields&nbsp;<math alttext=\"604 equals a left parenthesis 1 plus b right parenthesis Superscript 0\"><mn>604</mn><mo>=</mo><mi>a</mi><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>b</mi></mrow></mfenced><mn>0</mn></msup></math>, which is equivalent to&nbsp;<math alttext=\"604 equals a left parenthesis 1 right parenthesis\"><mn>604</mn><mo>=</mo><mi>a</mi><mfenced><mn>1</mn></mfenced></math>, or&nbsp;<math alttext=\"604 equals a\"><mn>604</mn><mo>=</mo><mi>a</mi></math>. Substituting <math alttext=\"604\"><mn>604</mn>\n</math> for <math alttext=\"a\"><mi>a</mi>\n</math> in the equation&nbsp;<math alttext=\"n equals a left parenthesis 1 plus b right parenthesis Superscript t\"><mi>n</mi><mo>=</mo><mi>a</mi><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>b</mi></mrow></mfenced><mi>t</mi></msup></math> yields&nbsp;<math alttext=\"n equals 604 left parenthesis 1 plus b right parenthesis Superscript t\"><mi>n</mi><mo>=</mo><mn>604</mn><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>b</mi></mrow></mfenced><mi>t</mi></msup></math>. The table also shows 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 606.42\"><mrow>\n\t<mi>n</mi>\n\t<mo>=</mo>\n\t<mn>606.42</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=\"606.42\"><mn>606.42</mn>\n</math> for <math alttext=\"n\"><mi>n</mi>\n</math> in the equation&nbsp;<math alttext=\"n equals 604 left parenthesis 1 plus b right parenthesis Superscript t\"><mi>n</mi><mo>=</mo><mn>604</mn><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>b</mi></mrow></mfenced><mi>t</mi></msup></math> yields&nbsp;<math alttext=\"606.42 equals 604 left parenthesis 1 plus b right parenthesis Superscript 1\"><mn>606.42</mn><mo>=</mo><mn>604</mn><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>b</mi></mrow></mfenced><mn>1</mn></msup></math>, or&nbsp;<math alttext=\"606.42 equals 604 left parenthesis 1 plus b right parenthesis\"><mn>606.42</mn><mo>=</mo><mn>604</mn><mfenced><mrow><mn>1</mn><mo>+</mo><mi>b</mi></mrow></mfenced></math>. Dividing both sides of this equation by <math alttext=\"604\"><mn>604</mn>\n</math> yields approximately <math alttext=\"1.004 equals 1 plus b\"><mn>1.004</mn><mo>=</mo><mn>1</mn><mo>+</mo><mi>b</mi></math>. Subtracting <math alttext=\"1\"><mn>1</mn>\n</math> from both sides of this equation yields that the value of <math alttext=\"b\"><mi>b</mi>\n</math> is approximately <math alttext=\"0.004\"><mn>0.004</mn>\n</math>. Substituting <math alttext=\"0.004\"><mn>0.004</mn>\n</math> for <math alttext=\"b\"><mi>b</mi>\n</math> in the equation <math alttext=\"n equals 604 left parenthesis 1 plus b right parenthesis Superscript t\"><mi>n</mi><mo>=</mo><mn>604</mn><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>b</mi></mrow></mfenced><mi>t</mi></msup></math> yields&nbsp;<math alttext=\"n equals 604 left parenthesis 1 plus 0.004 right parenthesis Superscript t\"><mi>n</mi><mo>=</mo><mn>604</mn><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mn>0.004</mn></mrow></mfenced><mi>t</mi></msup></math>. Therefore, of the choices, choice C best represents the relationship between <math alttext=\"t\"><mi>t</mi>\n</math> and <math alttext=\"n\"><mi>n</mi>\n</math>.</p>\n<p style=\"text-align: left;\">Choice A is incorrect and may result from conceptual or calculation errors.</p>\n<p style=\"text-align: left;\">Choice B is incorrect and may result from conceptual or calculation errors.</p>\n<p style=\"text-align: left;\">Choice D is incorrect and may result from conceptual or calculation errors.</p>","correct_answer":["C"],"has_media":false,"has_mathml":true,"has_table":true,"source_created_at":"2023-08-02T20:25:59.828000+00:00","source_updated_at":"2023-08-02T20:25:59.828000+00:00","options":[{"label":"A","content_html":"<p><math alttext=\"|n equals left parenthesis 1 plus 604 right parenthesis Superscript t|\"><mi>n</mi><mo>=</mo><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mn>604</mn></mrow></mfenced><mi>t</mi></msup></math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"|n equals left parenthesis 1 plus 0.004 right parenthesis Superscript t|\"><mi>n</mi><mo>=</mo><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mn>0.004</mn></mrow></mfenced><mi>t</mi></msup></math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"|n equals 604 left parenthesis 1 plus 0.004 right parenthesis Superscript t|\"><mi>n</mi><mo>=</mo><mn>604</mn><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mn>0.004</mn></mrow></mfenced><mi>t</mi></msup></math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"|n equals 0.004 left parenthesis 1 plus 604 right parenthesis Superscript t|\"><mi>n</mi><mo>=</mo><mn>0.004</mn><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mn>604</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}}