{"id":"cb_question_bank:5c0d8961-148f-49c1-907a-cd37148df0bc","source_id":"cb_question_bank","source_item_id":"5c0d8961-148f-49c1-907a-cd37148df0bc","external_id":"821bb037-c906-433b-a30b-6197a4337fb6","ibn":null,"question_id":"b7c74b73","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":"f left parenthesis x right parenthesis equals 5,470 left parenthesis 0.64 right parenthesis Superscript StartFraction x Over 12 EndFraction The function f gives the value, in dollars, of a certain piece of equipment after x months of use. If the value of the equipment decreases each year by p percent sign of its value the preceding year, what is the value of p ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: center;\"><math alttext=\"f left parenthesis x right parenthesis equals 5,470 left parenthesis 0.64 right parenthesis Superscript StartFraction x Over 12 EndFraction\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mrow><mn>5,470</mn></mrow><msup><mfenced><mrow><mn>0.64</mn></mrow></mfenced><mfrac><mi>x</mi><mn>12</mn></mfrac></msup></math></p>\n<p style=\"text-align: left;\">The function <math alttext=\"f\"><mi>f</mi>\n</math> gives the value, in dollars, of a certain piece of equipment after <math alttext=\"x\"><mi>x</mi>\n</math> months of use. If the value of the equipment decreases each <span style=\"text-decoration: underline;\">year</span> by&nbsp;<math alttext=\"p percent sign\"><mi>p</mi><mo>%</mo></math> of its value the preceding year, what is the value of <math alttext=\"p\"><mi>p</mi>\n</math>?</p>","rationale_html":"<p style=\"text-align: left;\">Choice C is correct. For a function of the form <math alttext=\"f left parenthesis x right parenthesis equals a left parenthesis r right parenthesis Superscript StartFraction x Over k EndFraction\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>a</mi><msup><mfenced><mi>r</mi></mfenced><mfrac><mi>x</mi><mi>k</mi></mfrac></msup></math>, where <math alttext=\"a\"><mi>a</mi>\n</math>, <math alttext=\"r\"><mi>r</mi>\n</math>, and <math alttext=\"k\"><mi>k</mi>\n</math> are constants and <math alttext=\"r less than 1\"><mi>r</mi><mo>&lt;</mo><mn>1</mn></math>, the value of <math alttext=\"f left parenthesis x right parenthesis\"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> decreases by <math alttext=\"100 left parenthesis 1 minus r right parenthesis percent sign\"><mn>100</mn><mfenced><mrow><mn>1</mn><mo>-</mo><mi>r</mi></mrow></mfenced><mo>%</mo></math> for every increase of <math alttext=\"x\"><mi>x</mi>\n</math> by <math alttext=\"k\"><mi>k</mi>\n</math>. In the given function, <math alttext=\"a equals 5,470\"><mrow>\n\t<mi>a</mi>\n\t<mo>=</mo>\n\t<mn>5,470</mn>\n</mrow>\n</math>, <math alttext=\"r equals 0.64\"><mrow>\n\t<mi>r</mi>\n\t<mo>=</mo>\n\t<mn>0.64</mn>\n</mrow>\n</math>, and <math alttext=\"k equals 12\"><mrow>\n\t<mi>k</mi>\n\t<mo>=</mo>\n\t<mn>12</mn>\n</mrow>\n</math>. Therefore, for the given function, the value of&nbsp;<math alttext=\"f left parenthesis x right parenthesis\"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> decreases by <math alttext=\"100 left parenthesis 1 minus 0.64 right parenthesis percent sign\"><mn>100</mn><mfenced><mrow><mn>1</mn><mo>-</mo><mn>0.64</mn></mrow></mfenced><mo>%</mo></math>, or&nbsp;<math alttext=\"36 percent sign\"><mn>36</mn><mo>%</mo></math>, for every increase of <math alttext=\"x\"><mi>x</mi>\n</math> by <math alttext=\"12\"><mn>12</mn>\n</math>. Since&nbsp;<math alttext=\"f left parenthesis x right parenthesis\"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> represents the value, in dollars, of the equipment after <math alttext=\"x\"><mi>x</mi>\n</math> months of use, it follows that the value of the equipment decreases every <math alttext=\"12\"><mn>12</mn>\n</math> months by&nbsp;<math alttext=\"36 percent sign\"><mn>36</mn><mo>%</mo></math> of its value the preceding <math alttext=\"12\"><mn>12</mn>\n</math> months. Since there are <math alttext=\"12\"><mn>12</mn>\n</math> months in a year, the value of the equipment decreases each year by&nbsp;<math alttext=\"36 percent sign\"><mn>36</mn><mo>%</mo></math> of its value the preceding year. Thus, the value of <math alttext=\"p\"><mi>p</mi>\n</math> is <math alttext=\"36\"><mn>36</mn>\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":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=\"4\"><mn>4</mn>\n</math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"5\"><mn>5</mn>\n</math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"36\"><mn>36</mn>\n</math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"64\"><mn>64</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}}