{"id":"cb_question_bank:caa574d3-4857-4ebe-843e-8bd1a803b148","source_id":"cb_question_bank","source_item_id":"caa574d3-4857-4ebe-843e-8bd1a803b148","external_id":null,"ibn":"026162-DC","question_id":"4b642eef","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.C.","skill":"Nonlinear functions","difficulty":"H","score_band":7,"answer_type":"spr","answer_source":"rationale","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"The total distance d , in meters, traveled by an object moving in a straight line can be modeled by a quadratic function that is defined in terms of t , where t is the time in seconds. At a time of 10.0 seconds, the total distance traveled by the object is 50.0 meters, and at a time of 20.0 seconds, the total distance traveled by the object is 200.0 meters. If the object was at a distance of 0 meters when t equals 0 , then what is the total distance traveled, in meters, by the object after 30.0 seconds?","stimulus":"","stimulus_html":null,"stem_html":"<p class=\"stem_paragraph \">The total distance <span class=\"italic\">d</span>, in meters, traveled by an object moving in a straight line can be modeled by a quadratic function that is defined in terms of <span class=\"italic\">t</span>, where <span class=\"italic\">t</span> is the time in seconds. At a time of 10.0 seconds, the total distance traveled by the object is 50.0 meters, and at a time of 20.0 seconds, the total distance traveled by the object is 200.0 meters. If the object was at a distance of 0 meters when <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/03be3f9ca86e73f91aae3da5f4c70ddbd7015f7b2182c3cada4a3b84a54d9546\" alt=\"t equals 0\"></span></span>, then what is the total distance traveled, in meters, by the object after 30.0 seconds?</p>\n","rationale_html":"<p>The correct answer is 450. The quadratic equation that models this situation can be written in the form <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/f8d3b34ac9e6f00fb09c2adc4689a331f906e769cf0532daf899a9defe66eddc\" alt=\"d equals, a, t squared, plus b t, plus c\"></span>, where <span class=\"italic\">a</span>, <span class=\"italic\">b</span>, and <span class=\"italic\">c</span> are constants. It’s given that the distance, <span class=\"italic\">d</span>, the object traveled was 0 meters when <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/03be3f9ca86e73f91aae3da5f4c70ddbd7015f7b2182c3cada4a3b84a54d9546\" alt=\"t equals 0 \"></span> seconds. These values can be substituted into the equation to solve for <span class=\"italic\">a</span>, <span class=\"italic\">b</span>, and <span class=\"italic\">c</span>: <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/0a2629a71502fcdbda5cf59ad0fea5fdf729eb748b9eaf351cd771d7379c9ad9\" alt=\"0 equals, a, times 0 squared, plus, b times 0, plus c\"></span>. Therefore, <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/d544e9591dbfd54ba320c077d21b86cfc11c2a940b4b1d6ab04c4f4a02d8cae4\" alt=\"c equals 0\"></span>, and it follows that <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/c140dc4d9cf0aa297b460b18c688a09970deacaa20fa6d7f3663c142ac229f6a\" alt=\"d equals, a, t squared, plus b t\"></span>. Since it’s also given that <span class=\"italic\">d</span> is 50 when <span class=\"italic\">t</span> is 10 and <span class=\"italic\">d</span> is 200 when <span class=\"italic\">t</span> is 20, these values for <span class=\"italic\">d</span> and <span class=\"italic\">t</span> can be substituted to create a system of two linear equations: <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/cd4a569dbbf81e4fb739f17e0b56835a2a36a05dfe22f2c518f43a2119cc2743\" alt=\"50 equals, a, times 10 squared, plus, b times 10 \"></span> and <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/77c137605bbc392c1ee94606b08326b9dbb35cabc8fad11ae72a8ff667690513\" alt=\"200 equals, a, times 20 squared, plus, b times 20\"></span>, or <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/be22c5bbfa157705c96c909fdb4f74563fc8efbe181bdaa38fc235b0fd45df69\" alt=\"10 a, plus b, equals 5 \"></span> and <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/b024867ef0a7fdf7ec46f8fccbd09d8e84b41bb57b3a82ef95ad1242ff0be83b\" alt=\"20 a, plus b, equals 10\"></span>. Subtracting the first equation from the second equation yields <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/d1677c63c080aed7903f6bc318d9a8eabb70f04e71396aa587663be1180e2c43\" alt=\"10 a, equals 5\"></span>, or <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/7cfda5bcc014a626b134da8a190436c1d8ec343f27599ff1527f9004a877e063\" alt=\"a, equals one half\"></span>. Substituting <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/acd304db4db3ee4a427462753769f86b9941b320104a3856da731c39db5d894c\" alt=\"one half\"></span> for <span class=\"italic\">a</span> in the first equation and solving for <span class=\"italic\">b</span> yields <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/93136d946c2e309cdb60ee98acd80bb8b661b2eb7da19101972f753e2b813320\" alt=\"b equals 0\"></span>. Therefore, the equation that represents this situation is <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/3293186b3faa8e5d1545890bb7566b372372b55ca4f2ac650ad6a0cff4ad21a7\" alt=\"d equals, one half t squared\"></span>. Evaluating this function when <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/8d45198ff5a95eb42020d8657da77a83262efcef7a79d46dc6726c27d95bbd90\" alt=\"t equals 30 \"></span> seconds yields <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/277d0ddfeb39c85a09b6378d806433e1d37e485c985642b4d17548238d3ad9cb\" alt=\"d, equals, one half open parenthesis, 30, close parenthesis, squared, equals 450\"></span>, or <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/bc61dead363696caf5b8a8c269a73446b86c954c255286fe6843457ab49ca857\" alt=\"d equals 450 \"></span> meters.</p>\n","correct_answer":["450"],"has_media":true,"has_mathml":false,"has_table":false,"source_created_at":"2023-08-02T20:25:59.620000+00:00","source_updated_at":"2023-08-02T20:25:59.620000+00:00","options":[],"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}}