{"id":182055,"date":"2025-01-13T09:41:52","date_gmt":"2025-01-13T09:41:52","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=182055"},"modified":"2025-01-13T09:41:54","modified_gmt":"2025-01-13T09:41:54","slug":"determine-the-profit-maximizing-output-for-a-firm-exhibiting-the-following","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/01\/13\/determine-the-profit-maximizing-output-for-a-firm-exhibiting-the-following\/","title":{"rendered":"Determine the profit maximizing output for a firm exhibiting the following"},"content":{"rendered":"\n
    \n
  1. Determine the profit maximizing output for a firm exhibiting the following:<\/li>\n<\/ol>\n\n\n\n

    Demand Schedule: P = 4000 \u2013 33Q<\/p>\n\n\n\n

    TC = 2Q3 \u2013 3Q2 + 400Q + 5000<\/p>\n\n\n\n

    The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n

    To determine the profit-maximizing output, we need to find the quantity (Q) at which the firm’s marginal revenue (MR) equals marginal cost (MC). Here’s the step-by-step process:<\/p>\n\n\n\n

    Step 1: Determine the Revenue Function<\/h3>\n\n\n\n

    The revenue (R) is given by price (P) times quantity (Q):
    [
    R = P \\times Q
    ]
    From the demand schedule ( P = 4000 – 33Q ), we substitute it into the revenue formula:
    [
    R = (4000 – 33Q) \\times Q = 4000Q – 33Q^2
    ]<\/p>\n\n\n\n

    Step 2: Determine Marginal Revenue (MR)<\/h3>\n\n\n\n

    Marginal revenue is the derivative of the revenue function with respect to Q:
    [
    MR = \\frac{dR}{dQ} = 4000 – 66Q
    ]<\/p>\n\n\n\n

    Step 3: Determine the Marginal Cost (MC)<\/h3>\n\n\n\n

    The total cost (TC) function is given as:
    [
    TC = 2Q^3 – 3Q^2 + 400Q + 5000
    ]
    Marginal cost is the derivative of the total cost function with respect to Q:
    [
    MC = \\frac{dTC}{dQ} = 6Q^2 – 6Q + 400
    ]<\/p>\n\n\n\n

    Step 4: Set MR Equal to MC<\/h3>\n\n\n\n

    To find the profit-maximizing output, set MR equal to MC:
    [
    4000 – 66Q = 6Q^2 – 6Q + 400
    ]
    Rearrange the equation:
    [
    6Q^2 + 60Q – 3600 = 0
    ]<\/p>\n\n\n\n

    Step 5: Solve the Quadratic Equation<\/h3>\n\n\n\n

    Solve the quadratic equation using the quadratic formula:
    [
    Q = \\frac{-b \\pm \\sqrt{b^2 – 4ac}}{2a}
    ]
    Where ( a = 6 ), ( b = 60 ), and ( c = -3600 ):
    [
    Q = \\frac{-60 \\pm \\sqrt{60^2 – 4(6)(-3600)}}{2(6)}
    ]
    [
    Q = \\frac{-60 \\pm \\sqrt{3600 + 86400}}{12}
    ]
    [
    Q = \\frac{-60 \\pm \\sqrt{90000}}{12}
    ]
    [
    Q = \\frac{-60 \\pm 300}{12}
    ]
    Thus, the two possible solutions for Q are:
    [
    Q = \\frac{240}{12} = 20 \\quad \\text{or} \\quad Q = \\frac{-360}{12} = -30
    ]
    Since quantity cannot be negative, the profit-maximizing output is ( Q = 20 ).<\/p>\n\n\n\n

    Step 6: Confirm the Profit-Maximizing Output<\/h3>\n\n\n\n

    At ( Q = 20 ), we can check that MR equals MC to confirm:
    [
    MR = 4000 – 66(20) = 4000 – 1320 = 2680
    ]
    [
    MC = 6(20)^2 – 6(20) + 400 = 6(400) – 120 + 400 = 2400 – 120 + 400 = 2680
    ]
    Since MR equals MC at ( Q = 20 ), this is the profit-maximizing output.<\/p>\n\n\n\n

    Conclusion<\/h3>\n\n\n\n

    The profit-maximizing output for the firm is 20 units. At this output, the firm’s marginal revenue equals its marginal cost, ensuring maximum profit.<\/p>\n","protected":false},"excerpt":{"rendered":"

    Demand Schedule: P = 4000 \u2013 33Q TC = 2Q3 \u2013 3Q2 + 400Q + 5000 The Correct Answer and Explanation is : To determine the profit-maximizing output, we need to find the quantity (Q) at which the firm’s marginal revenue (MR) equals marginal cost (MC). Here’s the step-by-step process: Step 1: Determine the Revenue […]<\/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-182055","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/182055","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=182055"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/182055\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=182055"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=182055"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=182055"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}