{"id":"cb_question_bank:daec4b85-07d3-4e5a-9ad1-86315ce951a8","source_id":"cb_question_bank","source_item_id":"daec4b85-07d3-4e5a-9ad1-86315ce951a8","external_id":null,"ibn":"10570-DC","question_id":"90bd9ef8","program":"SAT","module":"math","domain_code":"H","domain":"Algebra","skill_code":"H.E.","skill":"Linear inequalities in one or two variables","difficulty":"M","score_band":5,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"The average annual energy cost for a certain home is $4,334. The homeowner plans to spend $25,000 to install a geothermal heating system. The homeowner estimates that the average annual energy cost will then be $2,712. Which of the following inequalities can be solved to find t , the number of years after installation at which the total amount of energy cost savings will exceed the installation cost?","stimulus":"","stimulus_html":null,"stem_html":"<p class=\"stem_paragraph \">The average annual energy cost for a certain home is $4,334. The homeowner plans to spend $25,000 to install a geothermal heating system. The homeowner estimates that the average annual energy cost will then be $2,712. Which of the following inequalities can be solved to find <span class=\"italic\">t</span>, the number of years after installation at which the total amount of energy cost savings will exceed the installation&nbsp;cost?</p>\n","rationale_html":"<p>Choice B is correct. The savings each year from installing the geothermal heating system will be the average annual energy cost for the home before the geothermal heating system installation minus the average annual energy cost after the geothermal heating system installation, which is <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/0c65756e90bc0128d0d6d776dd88c7a6cd54cdee8d40c7e65be9bc694b4040f0\" alt=\"4,334 minus 2,712\"></span> dollars. In <span class=\"italic\">t</span> years, the savings will be <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/f31e859fac3ff0436b52dfd01a58c32a160bbadcefaacf16c52dba9d83ac1b89\" alt=\"open parenthesis, 4,334 minus 2,712, close parenthesis, times t\"></span> dollars. Therefore, the inequality that can be solved to find the number of years after installation at which the total amount of energy cost savings will exceed (be greater than) the installation cost, $25,000, is <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/f597da2377700130a7443aea6eeee8c3ccbf6ea4841417cf0bd44ddcccb76a94\" alt=\"25,000 is less than, open parenthesis, 4,334 minus 2,712, close parenthesis, times t\"></span>.</p><p>Choice A is incorrect. It gives the number of years after installation at which the total amount of energy cost savings will be less than the installation cost. Choice C is incorrect and may result from subtracting the average annual energy cost for the home from the onetime cost<br>\nof the geothermal heating system installation. To find the predicted total savings, the predicted average cost should be subtracted from the average annual energy cost before the installation, and the result should be multiplied by the number of years, <span class=\"italic\">t</span>. Choice D is incorrect and may result from misunderstanding the context. The ratio <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/da76251bc12f809860978d983318687c23ba536faf16a0ebddc95e9a4aaab740\" alt=\"4,332 over 2,712\"></span> compares the average energy cost before installation and the average energy cost after installation; it does not represent the savings.</p><p></p>\n","correct_answer":["B"],"has_media":true,"has_mathml":false,"has_table":false,"source_created_at":"2023-08-02T20:25:59.637000+00:00","source_updated_at":"2023-08-02T20:25:59.637000+00:00","options":[{"label":"A","content_html":"<span class=\"math_expression \">\n            <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/2a6a7f6ca6a0a8db8c3e09444a2f49b430c96338fbef991408856f1e8d49e3ab\" alt=\"25,000 is greater than, open parenthesis, 4,334 minus 2,712, close parenthesis, times t\"></span></span>\n","ord":0},{"label":"B","content_html":"<span class=\"math_expression \">\n            <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/472ae45fcec5117a62b50edc90e38ed5c95678c8af7f3b8469420dee8da17efd\" alt=\"25,000 is less than, open parenthesis, 4,334 minus 2,712, close parenthesis, times t\"></span></span>\n","ord":1},{"label":"C","content_html":"<span class=\"math_expression \">\n            <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/8dfe686641dedda2afdb7c5a585777c34fde88ad3f24505334503d28d2ca1318\" alt=\"25,000 minus 4,334, is greater than 2,712 t\"></span></span>\n","ord":2},{"label":"D","content_html":"<span class=\"math_expression \">\n            <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/598f352f9c302db7a4d03103969adbb79319060c57b0c97e4f37f25f41589005\" alt=\"25,000 is greater than the fraction with numerator 4,332, and denominator 2,712, end fraction, t\"></span></span>\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}}