{"id":"cb_question_bank:bae11d60-8f7a-4836-a14e-4f3ce02aea7a","source_id":"cb_question_bank","source_item_id":"bae11d60-8f7a-4836-a14e-4f3ce02aea7a","external_id":"ad9d7217-a0fd-419c-b72d-02d82ef7e7c4","ibn":null,"question_id":"84e8cc72","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 quadratic function models the height, in feet, of an object above the ground in terms of the time, in seconds, after the object is launched off an elevated surface. The model indicates the object has an initial height of 10 feet above the ground and reaches its maximum height of 1,034 feet above the ground 8 seconds after being launched. Based on the model, what is the height, in feet, of the object above the ground 10 seconds after being launched?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">A quadratic function models the height, in feet, of an object above the ground in terms of the time, in seconds, after the object is launched off an elevated surface. The model indicates the object has an initial height of <math alttext=\"10\"><mn>10</mn>\n</math> feet above the ground and reaches its maximum height of <math alttext=\"1,034\"><mn>1,034</mn>\n</math> feet above the ground <math alttext=\"8\"><mn>8</mn>\n</math> seconds after being launched. Based on the model, what is the height, in feet, of the object above the ground <math alttext=\"10\"><mn>10</mn>\n</math> seconds after being launched?</p>","rationale_html":"<p style=\"text-align: left;\">Choice C is correct. It's given that a quadratic function models the height, in feet, of an object above the ground in terms of the time, in seconds, after the object is launched off an elevated surface. This quadratic function can be defined by an equation of the form <math alttext=\"f left parenthesis x right parenthesis equals a left parenthesis x minus h right parenthesis squared plus k\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>a</mi><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>h</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mi>k</mi></math>, where <math alttext=\"f left parenthesis x right parenthesis\"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> is the height of the object <math alttext=\"x\"><mi>x</mi>\n</math> seconds after it was launched, and <math alttext=\"a\"><mi>a</mi>\n</math>, <math alttext=\"h\"><mi>h</mi>\n</math>, and <math alttext=\"k\"><mi>k</mi>\n</math> are constants such that the function reaches its maximum value, <math alttext=\"k\"><mi>k</mi>\n</math>, when <math alttext=\"x equals h\"><mrow>\n\t<mi>x</mi>\n\t<mo>=</mo>\n\t<mi>h</mi>\n</mrow>\n</math>. Since the model indicates the object reaches its maximum height of <math alttext=\"1,034\"><mn>1,034</mn>\n</math> feet above the ground <math alttext=\"8\"><mn>8</mn>\n</math> seconds after being launched, <math alttext=\"f left parenthesis x right parenthesis\"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> reaches its maximum value, <math alttext=\"1,034\"><mn>1,034</mn>\n</math>, when <math alttext=\"x equals 8\"><mrow>\n\t<mi>x</mi>\n\t<mo>=</mo>\n\t<mn>8</mn>\n</mrow>\n</math>. Therefore, <math alttext=\"k equals 1,034\"><mrow>\n\t<mi>k</mi>\n\t<mo>=</mo>\n\t<mn>1,034</mn>\n</mrow>\n</math> and <math alttext=\"h equals 8\"><mrow>\n\t<mi>h</mi>\n\t<mo>=</mo>\n\t<mn>8</mn>\n</mrow>\n</math>. Substituting <math alttext=\"8\"><mn>8</mn>\n</math> for <math alttext=\"h\"><mi>h</mi>\n</math> and <math alttext=\"1,034\"><mn>1,034</mn>\n</math> for <math alttext=\"k\"><mi>k</mi>\n</math> in the function <math alttext=\"f left parenthesis x right parenthesis equals a left parenthesis x minus h right parenthesis squared plus k\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>a</mi><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>h</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mi>k</mi></math> yields <math alttext=\"f left parenthesis x right parenthesis equals a left parenthesis x minus 8 right parenthesis squared plus 1,034\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>a</mi><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>8</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>1,034</mn></math>. Since the model indicates the object has an initial height of <math alttext=\"10\"><mn>10</mn>\n</math> feet above the ground, the value of <math alttext=\"f left parenthesis x right parenthesis\"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> is <math alttext=\"10\"><mn>10</mn>\n</math> when <math alttext=\"x equals 0\"><mrow>\n\t<mi>x</mi>\n\t<mo>=</mo>\n\t<mn>0</mn>\n</mrow>\n</math>. Substituting <math alttext=\"0\"><mn>0</mn>\n</math> for <math alttext=\"x\"><mi>x</mi>\n</math> and <math alttext=\"10\"><mn>10</mn>\n</math> for <math alttext=\"f left parenthesis x right parenthesis\"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> in the equation <math alttext=\"f left parenthesis x right parenthesis equals a left parenthesis x minus 8 right parenthesis squared plus 1,034\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>a</mi><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>8</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>1,034</mn></math> yields <math alttext=\"10 equals a left parenthesis 0 minus 8 right parenthesis squared plus 1,034\"><mn>10</mn><mo>=</mo><mi>a</mi><msup><mfenced><mrow><mn>0</mn><mo>-</mo><mn>8</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>1,034</mn></math>, or <math alttext=\"10 equals 64 a plus 1,034\"><mn>10</mn><mo>=</mo><mn>64</mn><mi>a</mi><mo>+</mo><mn>1,034</mn></math>. Subtracting <math alttext=\"1,034\"><mn>1,034</mn>\n</math> from both sides of this equation yields <math alttext=\"64 a equals negative 1,024\"><mn>64</mn><mi>a</mi><mo>=</mo><mo>-</mo><mn>1,024</mn></math>. Dividing both sides of this equation by <math alttext=\"64\"><mn>64</mn>\n</math> yields <math alttext=\"a equals negative 16\"><mrow>\n\t<mi>a</mi>\n\t<mo>=</mo>\n\t<mo>-</mo><mn>16</mn>\n</mrow>\n</math>. Therefore, the model can be represented by the equation <math alttext=\"f left parenthesis x right parenthesis equals minus 16 left parenthesis x minus 8 right parenthesis squared plus 1,034\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mn>16</mn><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>8</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>1,034</mn></math>. Substituting <math alttext=\"10\"><mn>10</mn>\n</math> for <math alttext=\"x\"><mi>x</mi>\n</math> in this equation yields <math alttext=\"f left parenthesis 10 right parenthesis equals minus 16 left parenthesis 10 minus 8 right parenthesis squared plus 1,034\"><mi>f</mi><mfenced><mn>10</mn></mfenced><mo>=</mo><mo>-</mo><mn>16</mn><msup><mfenced><mrow><mn>10</mn><mo>-</mo><mn>8</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>1,034</mn></math>, or <math alttext=\"f left parenthesis 10 right parenthesis equals 970\"><mi>f</mi><mfenced><mn>10</mn></mfenced><mo>=</mo><mn>970</mn></math>. Therefore, based on the model, <math alttext=\"10\"><mn>10</mn>\n</math> seconds after being launched, the height of the object above the ground is <math alttext=\"970\"><mn>970</mn>\n</math> feet.</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":"2023-08-02T20:25:59.825000+00:00","source_updated_at":"2023-08-02T20:25:59.825000+00:00","options":[{"label":"A","content_html":"<p><math alttext=\"234\"><mn>234</mn>\n</math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"778\"><mn>778</mn>\n</math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"970\"><mn>970</mn>\n</math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"1,014\"><mn>1,014</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}}