{"id":"cb_question_bank:20c2550c-c0bf-4676-b826-93010aee054c","source_id":"cb_question_bank","source_item_id":"20c2550c-c0bf-4676-b826-93010aee054c","external_id":"bf8e1345-911b-42f4-ad9c-688a5ce90cb6","ibn":null,"question_id":"7902bed0","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.C.","skill":"Nonlinear functions","difficulty":"H","score_band":6,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"A machine launches a softball from ground level. The softball reaches a maximum height of 51.84 meters above the ground at 1.8 seconds and hits the ground at 3.6 seconds. Which equation represents the height above ground h , in meters, of the softball t seconds after it is launched?","stimulus":"","stimulus_html":null,"stem_html":"

A machine launches a softball from ground level. The softball reaches a maximum height of 51.84\n meters above the ground at 1.8\n seconds and hits the ground at 3.6\n seconds. Which equation represents the height above ground h\n, in meters, of the softball t\n seconds after it is launched?

","rationale_html":"

Choice D is correct. An equation representing the height above ground h\n, in meters, of a softball t\n seconds after it is launched by a machine from ground level can be written in the form h=-at-b2+c, where a\n, b\n, and c\n are positive constants. In this equation, b\n represents the time, in seconds, at which the softball reaches its maximum height of c\n meters above the ground. It's given that this softball reaches a maximum height of 51.84\n meters above the ground at 1.8 seconds; therefore, b=1.8 and \n\tc\n\t=\n\t51.84\n\n. Substituting 1.8 for b\n and 51.84\n for c\n in the equation h=-at-b2+c yields h=-at-1.82+51.84. It's also given that this softball hits the ground at 3.6 seconds; therefore, h=0 when t=3.6. Substituting 0\n for h\n and 3.6 for t\n in the equation h=-at-1.82+51.84 yields 0=-a3.6-1.82+51.84, which is equivalent to 0=-a1.82+51.84, or 0=-3.24a+51.84. Adding \n\t3.24\n\ta\n\n to both sides of this equation yields 3.24a=51.84. Dividing both sides of this equation by 3.24\n yields a=16. Substituting 16\n for a\n in the equation h=-at-1.82+51.84 yields h=-16t-1.82+51.84. Therefore, h=-16t-1.82+51.84 represents the height above ground h\n, in meters, of this softball t\n seconds after it is launched.

\n

Choice A is incorrect. This equation represents a situation where the maximum height is 3.6 meters above the ground at 0\n seconds, not 51.84\n meters above the ground at 1.8 seconds.

\n

Choice B is incorrect. This equation represents a situation where the maximum height is 51.84\n meters above the ground at 0\n seconds, not 1.8 seconds.

\n

Choice C is incorrect and may result from conceptual or calculation errors.

","correct_answer":["D"],"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":"

\n\th\n\t=\n\t\n\t\t\n\t\t\t-\n\t\t\t\n\t\t\t\tt\n\t\t\t\t2\n\t\t\t\n\t\t\n\t\t+\n\t\t3.6\n\t\n\n

","ord":0},{"label":"B","content_html":"

\n\th\n\t=\n\t\n\t\t\n\t\t\t-\n\t\t\t\n\t\t\t\tt\n\t\t\t\t2\n\t\t\t\n\t\t\n\t\t+\n\t\t51.84\n\t\n\n

","ord":1},{"label":"C","content_html":"

\n\th\n\t=\n\t\n\t\t\n\t\t\t-\n\t\t\t16\n\t\t\t\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\tt\n\t\t\t\t\t\t-\n\t\t\t\t\t\t1.8\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t\t2\n\t\t\t\n\t\t\n\t\t-\n\t\t3.6\n\t\n\n

","ord":2},{"label":"D","content_html":"

\n\th\n\t=\n\t\n\t\t\n\t\t\t-\n\t\t\t16\n\t\t\t\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\tt\n\t\t\t\t\t\t-\n\t\t\t\t\t\t1.8\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t\t2\n\t\t\t\n\t\t\n\t\t+\n\t\t51.84\n\t\n\n

","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}}