{"id":"cb_question_bank:085f9e1d-b449-454e-9e38-6224fda4717c","source_id":"cb_question_bank","source_item_id":"085f9e1d-b449-454e-9e38-6224fda4717c","external_id":"1a8089dd-7f49-40bb-9cf4-9d9040daa84f","ibn":null,"question_id":"8c5e6702","program":"SAT","module":"math","domain_code":"H","domain":"Algebra","skill_code":"H.B.","skill":"Linear 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 window repair specialist charges dollar sign 220 for the first two hours of repair plus an hourly fee for each additional hour. The total cost for 5 hours of repair is dollar sign 400 . Which function f gives the total cost, in dollars, for x hours of repair, where x greater than or equals 2 ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">A window repair specialist charges <math alttext=\"dollar sign 220\"><mo>$</mo><mn>220</mn></math> for the first two hours of repair plus an hourly fee for each additional hour. The total cost for <math alttext=\"5\"><mn>5</mn>\n</math> hours of repair is <math alttext=\"dollar sign 400\"><mo>$</mo><mn>400</mn></math>. Which function <math alttext=\"f\"><mi>f</mi>\n</math> gives the total cost, in dollars, for <math alttext=\"x\"><mi>x</mi>\n</math> hours of repair, where <math alttext=\"x greater than or equals 2\"><mi>x</mi><mo>≥</mo><mn>2</mn></math>?</p>","rationale_html":"<p style=\"text-align: left;\">Choice A is correct. It’s given that the window repair specialist charges <math alttext=\"dollar sign 220\"><mo>$</mo><mn>220</mn></math> for the first two hours of repair plus an hourly fee for each additional hour. Let <math alttext=\"n\"><mi>n</mi>\n</math> represent the hourly fee for each additional hour after the first two hours. Since it’s given that <math alttext=\"x\"><mi>x</mi>\n</math> is the number of hours of repair, it follows that the charge generated by the hourly fee after the first two hours can be represented by the expression <math alttext=\"n left parenthesis x minus 2 right parenthesis\"><mrow>\n\t<mi>n</mi>\n\t<mfenced>\n\t\t<mrow>\n\t\t\t<mi>x</mi>\n\t\t\t<mo>-</mo>\n\t\t\t<mn>2</mn>\n\t\t</mrow>\n\t</mfenced>\n</mrow>\n</math>. Therefore, the total cost, in dollars, for <math alttext=\"x\"><mi>x</mi>\n</math> hours of repair is <math alttext=\"f left parenthesis x right parenthesis equals 220 plus n left parenthesis x minus 2 right parenthesis\"><mo>&nbsp;</mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>220</mn><mo>+</mo><mi>n</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfenced></math>.&nbsp;It’s given that the total cost for <math alttext=\"5\"><mn>5</mn>\n</math> hours of repair is <math alttext=\"dollar sign 400\"><mo>$</mo><mn>400</mn></math>. Substituting <math alttext=\"5\"><mn>5</mn>\n</math> for <math alttext=\"x\"><mi>x</mi>\n</math> and <math alttext=\"400\"><mn>400</mn>\n</math> for <math alttext=\"f left parenthesis x right parenthesis\"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> into the equation&nbsp;<math alttext=\"f left parenthesis x right parenthesis equals 220 plus n left parenthesis x minus 2 right parenthesis\"><mo>&nbsp;</mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>220</mn><mo>+</mo><mi>n</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfenced></math> yields <math alttext=\"400 equals 220 plus n left parenthesis 5 minus 2 right parenthesis\"><mn>400</mn><mo>=</mo><mn>220</mn><mo>+</mo><mi>n</mi><mfenced><mrow><mn>5</mn><mo>-</mo><mn>2</mn></mrow></mfenced></math>, or <math alttext=\"400 equals 220 plus 3 n\"><mn>400</mn><mo>=</mo><mn>220</mn><mo>+</mo><mn>3</mn><mi>n</mi></math>. Subtracting <math alttext=\"220\"><mn>220</mn>\n</math> from both sides of this equation yields <math alttext=\"180 equals 3 n\"><mrow>\n\t<mn>180</mn>\n\t<mo>=</mo>\n\t<mrow>\n\t\t<mn>3</mn>\n\t\t<mi>n</mi>\n\t</mrow>\n</mrow>\n</math>. Dividing both sides of this equation by <math alttext=\"3\"><mn>3</mn>\n</math> yields <math alttext=\"n equals 60\"><mrow>\n\t<mi>n</mi>\n\t<mo>=</mo>\n\t<mn>60</mn>\n</mrow>\n</math>. Substituting <math alttext=\"60\"><mn>60</mn>\n</math> for <math alttext=\"n\"><mi>n</mi>\n</math> in the equation <math alttext=\"f left parenthesis x right parenthesis equals 220 plus n left parenthesis x minus 2 right parenthesis\"><mo>&nbsp;</mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>220</mn><mo>+</mo><mi>n</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfenced></math> yields <math alttext=\"f left parenthesis x right parenthesis equals 220 plus 60 left parenthesis x minus 2 right parenthesis\"><mo>&nbsp;</mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>220</mn><mo>+</mo><mn>60</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfenced></math>, which is equivalent to <math alttext=\"f left parenthesis x right parenthesis equals 220 plus 60 x minus 120\"><mo>&nbsp;</mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>220</mn><mo>+</mo><mn>60</mn><mi>x</mi><mo>-</mo><mn>120</mn></math>, or <math alttext=\"f left parenthesis x right parenthesis equals 60 x plus 100\"><mo>&nbsp;</mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>60</mn><mi>x</mi><mo>+</mo><mn>100</mn></math>. Therefore, the total cost, in dollars, for <math alttext=\"x\"><mi>x</mi>\n</math> hours of repair is&nbsp;<math alttext=\"f left parenthesis x right parenthesis equals 60 x plus 100\"><mo>&nbsp;</mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>60</mn><mi>x</mi><mo>+</mo><mn>100</mn></math>.</p>\n<p style=\"text-align: left;\">Choice B is incorrect. This function represents the total cost, in dollars, for <math alttext=\"x\"><mi>x</mi>\n</math> hours of repair where the specialist charges <math alttext=\"dollar sign 340\"><mo>$</mo><mn>340</mn></math>, rather than <math alttext=\"dollar sign 220\"><mo>$</mo><mn>220</mn></math>, for the first two hours of repair.</p>\n<p style=\"text-align: left;\">Choice C is incorrect. This function represents the total cost, in dollars, for <math alttext=\"x\"><mi>x</mi>\n</math> hours of repair where the specialist charges <math alttext=\"dollar sign 160\"><mo>$</mo><mn>160</mn></math>, rather than <math alttext=\"dollar sign 220\"><mo>$</mo><mn>220</mn></math>, for the first two hours of repair, and an hourly fee of <math alttext=\"dollar sign 80\"><mo>$</mo><mn>80</mn></math>, rather than <math alttext=\"dollar sign 60\"><mo>$</mo><mn>60</mn></math>, after the first two hours.</p>\n<p style=\"text-align: left;\">Choice D is incorrect. This function represents the total cost, in dollars, for <math alttext=\"x\"><mi>x</mi>\n</math> hours of repair where the specialist charges <math alttext=\"dollar sign 380\"><mo>$</mo><mn>380</mn></math>, rather than <math alttext=\"dollar sign 220\"><mo>$</mo><mn>220</mn></math>, for the first two hours of repair, and an hourly fee of <math alttext=\"dollar sign 80\"><mo>$</mo><mn>80</mn></math>, rather than <math alttext=\"dollar sign 60\"><mo>$</mo><mn>60</mn></math>, after the first two hours.</p>","correct_answer":["A"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.810000+00:00","source_updated_at":"2023-08-02T20:25:59.810000+00:00","options":[{"label":"A","content_html":"<p><math alttext=\"f left parenthesis x right parenthesis equals 60 x plus 100\"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>60</mn><mi>x</mi><mo>+</mo><mn>100</mn></math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"f left parenthesis x right parenthesis equals 60 x plus 220\"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>60</mn><mi>x</mi><mo>+</mo><mn>220</mn></math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"f left parenthesis x right parenthesis equals 80 x\"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>80</mn><mi>x</mi></math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"f left parenthesis x right parenthesis equals 80 x plus 220\"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>80</mn><mi>x</mi><mo>+</mo><mn>220</mn></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}}