{"id":219453,"date":"2025-05-26T07:48:47","date_gmt":"2025-05-26T07:48:47","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=219453"},"modified":"2025-05-26T07:48:49","modified_gmt":"2025-05-26T07:48:49","slug":"solve-linear-programming-using-the-graphical-method","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/26\/solve-linear-programming-using-the-graphical-method\/","title":{"rendered":"Solve linear programming using the graphical method"},"content":{"rendered":"\n<p>Solve linear programming using the graphical method: a calculator company produces a scientific calculator and a graphing calculator. Long-term projections indicate an expected demand of at least 100 scientific and 80 graphing calculators each day. Because of limitations on production capacity, no more than 200 scientific and 170 graphing calculators can be made daily. To satisfy a shipping contract, a total of at least 200 calculators must be shipped each day. If each scientific calculator sold results in a $2 loss, but each graphing calculator produces a $5 profit, how many of each type should be made daily to maximize net profits?<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Correct Answer:<\/strong><\/h3>\n\n\n\n<p>Produce <strong>100 scientific calculators<\/strong> and <strong>170 graphing calculators<\/strong> daily.<br><strong>Maximum net profit = $650<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Explanation <\/strong><\/h3>\n\n\n\n<p>This is a linear programming problem with two decision variables:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Let <strong>x<\/strong> = number of scientific calculators produced per day<\/li>\n\n\n\n<li>Let <strong>y<\/strong> = number of graphing calculators produced per day<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Objective Function:<\/strong><\/h4>\n\n\n\n<p>We aim to <strong>maximize profit<\/strong>, given by:<br><strong>Profit (P) = -2x + 5y<\/strong><\/p>\n\n\n\n<p>(Each scientific calculator results in a $2 loss \u2192 -2x, each graphing calculator gives a $5 profit \u2192 +5y)<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Constraints:<\/strong><\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Demand constraints (minimum production):<\/strong>\n<ul class=\"wp-block-list\">\n<li>x \u2265 100 (at least 100 scientific calculators)<\/li>\n\n\n\n<li>y \u2265 80 (at least 80 graphing calculators)<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Capacity constraints (maximum production):<\/strong>\n<ul class=\"wp-block-list\">\n<li>x \u2264 200<\/li>\n\n\n\n<li>y \u2264 170<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Shipping constraint:<\/strong>\n<ul class=\"wp-block-list\">\n<li>x + y \u2265 200 (at least 200 calculators total)<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Feasible Region:<\/strong><\/h3>\n\n\n\n<p>Graph these constraints on the x-y plane and find the intersection area (feasible region). Check <strong>corner points<\/strong> of this region \u2014 these give potential optimal solutions.<\/p>\n\n\n\n<p>The relevant corner points (from graphical analysis) are:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>(100, 170)<\/li>\n\n\n\n<li>(200, 170)<\/li>\n\n\n\n<li>(200, 80)<\/li>\n\n\n\n<li>(120, 80) \u2014 intersection of x + y = 200 and y = 80<\/li>\n\n\n\n<li>(100, 100) \u2014 intersection of x + y = 200 and x = 100<\/li>\n<\/ol>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Evaluate the objective function at each:<\/strong><\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>(100, 170): P = -2(100) + 5(170) = -200 + 850 = <strong>650<\/strong> \u2705<\/li>\n\n\n\n<li>(200, 170): P = -400 + 850 = <strong>450<\/strong><\/li>\n\n\n\n<li>(200, 80): P = -400 + 400 = <strong>0<\/strong><\/li>\n\n\n\n<li>(120, 80): P = -240 + 400 = <strong>160<\/strong><\/li>\n\n\n\n<li>(100, 100): P = -200 + 500 = <strong>300<\/strong><\/li>\n<\/ol>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Conclusion:<\/strong><\/h3>\n\n\n\n<p>The <strong>maximum net profit is $650<\/strong>, achieved by producing <strong>100 scientific calculators<\/strong> and <strong>170 graphing calculators<\/strong> daily. This satisfies all constraints and gives the best outcome.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/05\/learnexams-banner6-28.jpeg\" alt=\"\" class=\"wp-image-219454\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Solve linear programming using the graphical method: a calculator company produces a scientific calculator and a graphing calculator. Long-term projections indicate an expected demand of at least 100 scientific and 80 graphing calculators each day. Because of limitations on production capacity, no more than 200 scientific and 170 graphing calculators can be made daily. To [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-219453","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/219453","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=219453"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/219453\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=219453"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=219453"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=219453"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}