{"id":587,"date":"2025-10-09T15:23:31","date_gmt":"2025-10-09T15:23:31","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=587"},"modified":"2026-04-13T16:39:37","modified_gmt":"2026-04-13T16:39:37","slug":"functions-of-several-variables-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/functions-of-several-variables-apply-it\/","title":{"raw":"Functions of Several Variables: Apply It","rendered":"Functions of Several Variables: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Recognize a function of two variables and identify its domain and range<\/li>\r\n \t<li>Sketch a graph of a function of two variables<\/li>\r\n \t<li>Analyze functions by looking at their level curves and traces<\/li>\r\n \t<li>Identify functions of three or more variables and understand their level surfaces<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Monitoring Air Quality in a Warehouse<\/h2>\r\nYou work as a safety coordinator at a large distribution warehouse. Air quality sensors throughout the building measure particulate concentration, which varies based on location due to ventilation patterns and equipment placement. Understanding how concentration levels change across the warehouse floor\u2014and eventually throughout the full three-dimensional space\u2014helps you determine where to install additional filtration units and place warning signage. The particulate concentration (measured in micrograms per cubic meter above the acceptable baseline) at any point on the warehouse floor is modeled by [latex]P(x, y) = 18 - 2x^2 - 2y^2[\/latex], where [latex]x[\/latex] and [latex]y[\/latex] are measured in tens of meters from the center of the building.\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]322640[\/ohm_question]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]322641[\/ohm_question]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]322642[\/ohm_question]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]322644[\/ohm_question]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]322646[\/ohm_question]\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Recognize a function of two variables and identify its domain and range<\/li>\n<li>Sketch a graph of a function of two variables<\/li>\n<li>Analyze functions by looking at their level curves and traces<\/li>\n<li>Identify functions of three or more variables and understand their level surfaces<\/li>\n<\/ul>\n<\/section>\n<h2>Monitoring Air Quality in a Warehouse<\/h2>\n<p>You work as a safety coordinator at a large distribution warehouse. Air quality sensors throughout the building measure particulate concentration, which varies based on location due to ventilation patterns and equipment placement. Understanding how concentration levels change across the warehouse floor\u2014and eventually throughout the full three-dimensional space\u2014helps you determine where to install additional filtration units and place warning signage. The particulate concentration (measured in micrograms per cubic meter above the acceptable baseline) at any point on the warehouse floor is modeled by [latex]P(x, y) = 18 - 2x^2 - 2y^2[\/latex], where [latex]x[\/latex] and [latex]y[\/latex] are measured in tens of meters from the center of the building.<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm322640\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=322640&theme=lumen&iframe_resize_id=ohm322640&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm322641\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=322641&theme=lumen&iframe_resize_id=ohm322641&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm322642\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=322642&theme=lumen&iframe_resize_id=ohm322642&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm322644\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=322644&theme=lumen&iframe_resize_id=ohm322644&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm322646\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=322646&theme=lumen&iframe_resize_id=ohm322646&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n","protected":false},"author":15,"menu_order":9,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-587","chapter","type-chapter","status-publish","hentry"],"part":426,"module-header":"apply_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/587","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":5,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/587\/revisions"}],"predecessor-version":[{"id":2854,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/587\/revisions\/2854"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/426"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/587\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=587"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=587"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=587"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=587"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}