{"id":"cb_question_bank:7924967f-60ae-4964-adad-cab59989eddf","source_id":"cb_question_bank","source_item_id":"7924967f-60ae-4964-adad-cab59989eddf","external_id":"e599f748-6055-4c3c-83f8-5366aad721b3","ibn":null,"question_id":"9da41c80","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.C.","skill":"Nonlinear functions","difficulty":"E","score_band":3,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"A ball is dropped from an initial height of 22 feet and bounces off the ground repeatedly. The function h estimates that the maximum height reached after each time the ball hits the ground is 85 percent sign of the maximum height reached after the previous time the ball hit the ground. Which equation defines h , where h left parenthesis n right parenthesis is the estimated maximum height of the ball after it has hit the ground n times and n is a whole number greater than 1 and less than 10 ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">A ball is dropped from an initial height of <math alttext=\"22\"><mn>22</mn>\n</math> feet and bounces off the ground repeatedly. The function <math alttext=\"h\"><mi>h</mi>\n</math> estimates that the maximum height reached after each time the ball hits the ground is <math alttext=\"85 percent sign\"><mn>85</mn>\n<mo>%</mo></math> of the maximum height reached after the previous time the ball hit the ground. Which equation defines <math alttext=\"h\"><mi>h</mi>\n</math>, where <math alttext=\"h left parenthesis n right parenthesis\"><mi>h</mi><mo>(</mo><mi>n</mi><mo>)</mo></math> is the estimated maximum height of the ball after it has hit the ground <math alttext=\"n\"><mi>n</mi>\n</math> times and <math alttext=\"n\"><mi>n</mi>\n</math> is a whole number greater than <math alttext=\"1\"><mn>1</mn>\n</math> and less than <math alttext=\"10\"><mn>10</mn>\n</math>?</p>","rationale_html":"<p style=\"text-align: left;\">Choice B is correct. It's given that for the function <math alttext=\"h\"><mi>h</mi>\n</math>, <math alttext=\"h left parenthesis n right parenthesis\"><mi>h</mi><mfenced><mi>n</mi></mfenced></math> is the estimated maximum height, in feet, of the ball after it has hit the ground <math alttext=\"n\"><mi>n</mi>\n</math> times. It's also given that the function <math alttext=\"h\"><mi>h</mi>\n</math> estimates that the maximum height reached after each time the ball hits the ground is <math alttext=\"85 percent sign\"><mn>85</mn><mo>%</mo></math> of the maximum height reached after the previous time the ball hit the ground. It follows that <math alttext=\"h\"><mi>h</mi>\n</math> is a decreasing exponential function that can be written in the form&nbsp;<math alttext=\"h left parenthesis n right parenthesis equals a left parenthesis StartFraction p Over 100 EndFraction right parenthesis Superscript n\"><mi>h</mi><mfenced><mi>n</mi></mfenced><mo>=</mo><mi>a</mi><msup><mfenced><mfrac><mi>p</mi><mn>100</mn></mfrac></mfenced><mi>n</mi></msup></math>, where <math alttext=\"a\"><mi>a</mi>\n</math> is the initial height, in feet, the ball was dropped from and the function estimates that the maximum height reached after each time the ball hits the ground is <math alttext=\"p percent sign\"><mi>p</mi><mo>%</mo></math> of the maximum height reached after the previous time the ball hit the ground. It's given that the ball is dropped from an initial height of <math alttext=\"22\"><mn>22</mn>\n</math> feet. Therefore, <math alttext=\"a equals 22\"><mrow>\n\t<mi>a</mi>\n\t<mo>=</mo>\n\t<mn>22</mn>\n</mrow>\n</math>. Since the function <math alttext=\"h\"><mi>h</mi>\n</math> estimates that the maximum height reached after each time the ball hits the ground is <math alttext=\"85 percent sign\"><mn>85</mn><mo>%</mo></math> of the maximum height reached after the previous time the ball hit the ground, <math alttext=\"p equals 85\"><mrow>\n\t<mi>p</mi>\n\t<mo>=</mo>\n\t<mn>85</mn>\n</mrow>\n</math>. Substituting <math alttext=\"22\"><mn>22</mn>\n</math> for <math alttext=\"a\"><mi>a</mi>\n</math> and <math alttext=\"85\"><mn>85</mn>\n</math> for <math alttext=\"p\"><mi>p</mi>\n</math> in the equation <math alttext=\"h left parenthesis n right parenthesis equals a left parenthesis StartFraction p Over 100 EndFraction right parenthesis Superscript n\"><mi>h</mi><mfenced><mi>n</mi></mfenced><mo>=</mo><mi>a</mi><msup><mfenced><mfrac><mi>p</mi><mn>100</mn></mfrac></mfenced><mi>n</mi></msup></math> yields <math alttext=\"h left parenthesis n right parenthesis equals 22 left parenthesis StartFraction 85 Over 100 EndFraction right parenthesis Superscript n\"><mi>h</mi><mfenced><mi>n</mi></mfenced><mo>=</mo><mn>22</mn><msup><mfenced><mfrac><mn>85</mn><mn>100</mn></mfrac></mfenced><mi>n</mi></msup></math>, or&nbsp;<math alttext=\"h left parenthesis n right parenthesis equals 22 left parenthesis 0.85 right parenthesis Superscript n\"><mi>h</mi><mfenced><mi>n</mi></mfenced><mo>=</mo><mn>22</mn><msup><mfenced><mn>0.85</mn></mfenced><mi>n</mi></msup></math>.</p>\n<p style=\"text-align: left;\">Choice A is incorrect. This function estimates that the maximum height reached after each time the ball hits the ground is&nbsp;<math alttext=\"22 percent sign\"><mn>22</mn><mo>%</mo></math>, not&nbsp;<math alttext=\"85 percent sign\"><mn>85</mn><mo>%</mo></math>, of the maximum height reached after the previous time the ball hit the ground.</p>\n<p style=\"text-align: left;\">Choice C is incorrect. This function estimates that the ball is dropped from an initial height of <math alttext=\"85\"><mn>85</mn>\n</math> feet, not <math alttext=\"22\"><mn>22</mn>\n</math> feet, and that the maximum height reached after each time the ball hits the ground is <math alttext=\"22 percent sign\"><mn>22</mn><mo>%</mo></math>, not <math alttext=\"85 percent sign\"><mn>85</mn><mo>%</mo></math>, of the maximum height reached after the previous time the ball hit the ground.</p>\n<p style=\"text-align: left;\">Choice D is incorrect. This function estimates that the ball is dropped from an initial height of <math alttext=\"85\"><mn>85</mn>\n</math> feet, not <math alttext=\"22\"><mn>22</mn>\n</math> feet.</p>","correct_answer":["B"],"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=\"h left parenthesis n right parenthesis equals 22 left parenthesis 0.22 right parenthesis Superscript n\"><mi>h</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mrow><mn>22</mn></mrow><msup><mrow><mo>(</mo><mrow><mn>0.22</mn></mrow><mo>)</mo></mrow><mi>n</mi></msup></math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"h left parenthesis n right parenthesis equals 22 left parenthesis 0.85 right parenthesis Superscript n\"><mi>h</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mrow><mn>22</mn></mrow><msup><mrow><mo>(</mo><mrow><mn>0.85</mn></mrow><mo>)</mo></mrow><mi>n</mi></msup></math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"h left parenthesis n right parenthesis equals 85 left parenthesis 0.22 right parenthesis Superscript n\"><mi>h</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mrow><mn>85</mn></mrow><msup><mfenced><mrow><mn>0.22</mn></mrow></mfenced><mi>n</mi></msup></math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"h left parenthesis n right parenthesis equals 85 left parenthesis 0.85 right parenthesis Superscript n\"><mi>h</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mrow><mn>85</mn></mrow><msup><mrow><mo>(</mo><mrow><mn>0.85</mn></mrow><mo>)</mo></mrow><mi>n</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}}