{"id":1039,"date":"2025-07-21T22:50:53","date_gmt":"2025-07-21T22:50:53","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1039"},"modified":"2026-01-14T19:38:21","modified_gmt":"2026-01-14T19:38:21","slug":"variation-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/variation-learn-it-3\/","title":{"raw":"Variation: Learn It 3","rendered":"Variation: Learn It 3"},"content":{"raw":"

Joint Variation<\/h2>\r\nMany situations are more complicated than a basic direct variation or inverse variation model. One variable often depends on multiple other variables. When a variable is dependent on the product or quotient of two or more variables, this is called joint variation<\/strong>. This concept extends the idea of direct variation to multiple variables and is often used in various scientific and engineering contexts.\r\n\r\n
\r\n

joint variation<\/h3>\r\nJoint variation occurs when a variable varies directly or inversely with multiple variables.\r\n\r\nFor instance:\r\n
    \r\n \t
  • If [latex]x[\/latex] varies directly with both [latex]y[\/latex] and [latex]z[\/latex], we have [latex]x=kyz[\/latex].<\/li>\r\n \t
  • If [latex]x[\/latex] varies directly with [latex]y[\/latex] and inversely with [latex]z[\/latex], we have [latex]x=\\dfrac{ky}{z}[\/latex].<\/li>\r\n<\/ul>\r\nNotice that we only use one constant in a joint variation equation.\r\n\r\n<\/section>
    A quantity [latex]x[\/latex]\u00a0varies directly with the square of [latex]y[\/latex]\u00a0and inversely with the cube root of [latex]z[\/latex]. If [latex]x=6[\/latex]\u00a0when [latex]y=2[\/latex]\u00a0and [latex]z=8[\/latex], find [latex]x[\/latex]\u00a0when [latex]y=1[\/latex]\u00a0and [latex]z=27[\/latex].[reveal-answer q=\"396823\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"396823\"]Begin by writing an equation to show the relationship between the variables.\r\n

    [latex]x=\\dfrac{k{y}^{2}}{\\sqrt[3]{z}}[\/latex]<\/p>\r\nSubstitute [latex]x=6[\/latex], [latex]y=2[\/latex], and [latex]z=8[\/latex]\u00a0to find the value of the constant [latex]k[\/latex].\r\n

    [latex]\\begin{align}6&=\\dfrac{k{2}^{2}}{\\sqrt[3]{8}} \\\\[1mm] 6&=\\dfrac{4k}{2} \\\\[1mm] 3&=k \\end{align}[\/latex]<\/p>\r\nNow we can substitute the value of the constant into the equation for the relationship.\r\n

    [latex]x=\\dfrac{3{y}^{2}}{\\sqrt[3]{z}}[\/latex]<\/p>\r\nTo find [latex]x[\/latex]\u00a0when [latex]y=1[\/latex]\u00a0and [latex]z=27[\/latex], we will substitute values for [latex]y[\/latex]\u00a0and [latex]z[\/latex]\u00a0into our equation.\r\n

    [latex]\\begin{align}x&=\\dfrac{3{\\left(1\\right)}^{2}}{\\sqrt[3]{27}} \\\\[1mm] &=1 \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>

    [ohm_question hide_question_numbers=1]318961[\/ohm_question]<\/section>
    [ohm_question hide_question_numbers=1]318962[\/ohm_question]<\/section>","rendered":"

    Joint Variation<\/h2>\n

    Many situations are more complicated than a basic direct variation or inverse variation model. One variable often depends on multiple other variables. When a variable is dependent on the product or quotient of two or more variables, this is called joint variation<\/strong>. This concept extends the idea of direct variation to multiple variables and is often used in various scientific and engineering contexts.<\/p>\n

    \n

    joint variation<\/h3>\n

    Joint variation occurs when a variable varies directly or inversely with multiple variables.<\/p>\n

    For instance:<\/p>\n

      \n
    • If [latex]x[\/latex] varies directly with both [latex]y[\/latex] and [latex]z[\/latex], we have [latex]x=kyz[\/latex].<\/li>\n
    • If [latex]x[\/latex] varies directly with [latex]y[\/latex] and inversely with [latex]z[\/latex], we have [latex]x=\\dfrac{ky}{z}[\/latex].<\/li>\n<\/ul>\n

      Notice that we only use one constant in a joint variation equation.<\/p>\n<\/section>\n

      A quantity [latex]x[\/latex]\u00a0varies directly with the square of [latex]y[\/latex]\u00a0and inversely with the cube root of [latex]z[\/latex]. If [latex]x=6[\/latex]\u00a0when [latex]y=2[\/latex]\u00a0and [latex]z=8[\/latex], find [latex]x[\/latex]\u00a0when [latex]y=1[\/latex]\u00a0and [latex]z=27[\/latex].<\/p>\n